Surface patch techniques for computational geometry

ABSTRACT

A method and system for computer aided design (CAD) is disclosed for designing geometric objects, wherein interpolation and/or blending between such objects is performed while deformation data is being input. Thus, a designer obtains immediate feedback to input modifications without separately entering a command(s) for performing such deformations. A novel N-sided surface generation technique is also disclosed herein to efficiently and accurately convert surfaces of high polynomial degree into a collection of lower degree surfaces. E.g., the N-sided surface generation technique disclosed herein subdivides parameter space objects (e.g., polygons) of seven or more sides into a collection of subpolygons, wherein each subpolygon has a reduced number of sides. More particularly, each subpolygon has 3 or 4 sides. The present disclosure is particularly useful for designing the shape of surfaces. Thus, the present disclosure is applicable to various design domains such as the design of, e.g., bottles, vehicles, and watercraft. Additionally, the present disclosure provides for efficient animation via repeatedly modifying surfaces of an animated object such as a representation of a face.

CROSS REFERENCE TO RELATED APPLICATIONS

The present application is a continuation of U.S. patent applicationSer. No. 15/043,880, filed Feb. 15, 2016, which is a continuation ofU.S. patent application Ser. No. 14/461,439 filed Aug. 17, 2014 (nowU.S. Pat. No. 9,262,859), which is a continuation of U.S. patentapplication Ser. No. 13/607,590 filed Sep. 7, 2012 (now U.S. Pat. No.8,836,701), which is a continuation of U.S. patent application Ser. No.12/460,587 filed Jul. 20, 2009, which claims the benefit of U.S.Provisional Patent Application Ser. No. 61/135,341 filed Jul. 18, 2008,and claims the benefit of U.S. Provisional Patent Application Ser. No.61/211,714 filed Apr. 1, 2009; all of the above-referenced applicationsare incorporated herein by reference in their entirety.

FIELD OF THE INVENTION

The present invention relates to a system and method for performingcomputer aided design, and, in particular, to efficient computationaltechniques for blending between representations of geometric objects,and more particularly to computer implemented methods for designingshapes.

BACKGROUND

A designer using a computer aided design (CAD) computational system willtypically approach the design of a free form geometric object (such as asurface) by first specifying prominent and/or necessary subportions ofthe geometric object through which the object is constrained to pass.Subsequently, a process is activated for generating the geometric objectthat conforms to the constraining subportions provided. In particular,such subportions may be points, curves, surfaces and/or higherdimensional geometric objects. For example, a designer that designs asurface may construct and position a plurality of curves through whichthe intended surface must pass (each such curve also being denotedherein as a “feature line” or “feature curve”). Thus, the intendedsurface is, in general, expected to have geometric characteristics (suchas differentiability and curvature) that, substantially, only change tothe extent necessary in order to satisfy the constraints placed upon thesurface by the plurality of curves. That is, the designer expects thegenerated surface to be what is typically referred to as “fair” by thoseskilled in the art. Thus, the designer typically constructs such featurecurves and positions them where the intended surface is likely to changeits geometric shape in a way that cannot be easily interpolated fromother subportions of the surface already designed.

As a more specific example, when designing containers such as bottles,an intended exterior surface of a bottle may be initially specified bysubportions such as: (a) feature curves positioned in high curvatureportions of the bottle surface, and (b) surface subareas havingparticular geometric characteristics such as having a shape or contourupon which a bottle label can be smoothly applied. Thus, the intentionof a bottle surface designer is to construct a bottle design thatsatisfies his/her input constraints and that is also fair. Moreover, thedesigner may desire to generate holes for handles, as well as, e.g.,ergonomic bottle grips by deforming various portions of the bottlesurface and still have the bottle surface fair.

There has heretofore, however, been no CAD system wherein a designer (ormore generally, user) of geometric objects can easily and efficientlyexpress his/her design intent by inputting constraints and having theresulting geometric object be fair. That is, the designer/user mayencounter lengthy delays due to substantial computational overheadand/or the designer/user may be confronted with non-intuitive geometricobject definition and deformation techniques that require substantialexperience to effectively use. For example, many prior art CAD systemsprovide techniques for allowing surfaces to be designed and/or deformedby defining and/or manipulating designated points denoted as “controlpoints.” However, such techniques can be computationally expensive,non-intuitive, and incapable of easily deforming more than a local areaof the surface associated with such a control point. Additionally, someprior art CAD systems provide techniques for defining and/or deformingsurfaces via certain individually designated control vectors. That is,the direction of these vectors may be used to define the shape orcontour of an associated surface. However, a designer's intent may noteasily correspond to a surface design technique using such controlvectors since each of the control vectors typically corresponds to onlya single point of the surface isolated from other surface points havingcorresponding control vectors. Thus, such techniques are, at most, onlyable to deform an area of the surface local to such points havingcorresponding control vectors.

Additionally, such prior art CAD systems may also have difficulties inprecisely performing blending and trimming operations. For example, twogeometric objects intended to abut one another along a common boundarymay not be within a insufficient tolerance to one another at theboundary. That is, there may be sufficiently large gaps between thegeometric objects that the boundary may not be considered “water tight,”which may be problematic in certain machining operations and otheroperations like Boolean operations on solids.

Prior art discloses techniques for generating multisided surface patchesfor geometric modeling. Such techniques are disclosed in U.S. Pat. No.7,236,167 filed Oct. 20, 2003 by Lee et. al. which is fully incorporatedherein by reference. Surfaces resulting from such techniques are able torepresent N-sided object space surfaces (also referred to as NSS herein)for substantially all reasonable values for the number N (e.g., N≤20),wherein N is the number of boundary curves of the resulting surface(which is sometimes referred to as a “patch”). As an illustrativeexample, FIG. 45 shows a pentagon 3009 in parametric space, wherein eachof its five sides is mapped into a corresponding portion of the boundaryof the surface 3011 in object space, and the interior of the pentagon3009 maps into the interior of the surface 3011 so that this interiorsmoothly meets this boundary. Moreover, various continuity constraintsand/or smooth transitions across the boundaries between two or more suchobject space patches 3011 may be imposed as' one skilled in the art willunderstand. Furthermore, there are various computational methods forexpressing NSS.

Assuming an NSS surface is represented by a polynomial (and accordingly,its boundary curves are also representable by polynomials); thepolynomial degree of NSS is important in that the higher degree, thegreater flexibility to generate, e.g., desired curves and/or contouredsurfaces. However, such flexibility comes at a price; e.g., geometricobjects generated from high degree polynomial mappings arecomputationally expensive to generate and manipulate. Additionally, suchhigh degree polynomial mappings can also affect the precision of variouscomputations, e.g., due to the accumulation of computational errors.Accordingly, many (if not most) commercially successful geometricmodelers limit the polynomial degree of the curves, surfaces and/orsolids that they generate and manipulate to be below a predeterminedvalue. For example, the geometric modeler by ACIS™ uses polynomialshaving a maximum of degree 25, and the geometric modeler by Rhino™ usespolynomials having a maximum of degree 32. Accordingly, when a geometricobject representation is desired to be imported into such a commercialgeometric modeler, it is very desirable for the object representation tohave as low a polynomial degree as possible, and certainly below themaximum polynomial degree for which the target geometric modeler hasbeen tested for reliability and accuracy. In particular, if thepolynomial degree of the geometric object is imported directly into sucha modeler, and the object's degree is above such the maximum recommendeddegree for the modeler, unexpected results may occur, e.g., the modelermay crash, an unacceptable build up computational errors may occur,and/or very lengthy processing times may be experienced, or even worse,undetected errors/inaccuracies may be incorporated into a geometricmodel generated by the modeler. Alternatively, prior to be imported, ageometric object having a representation of a too high degree may beconverted into, e.g., a plurality of lower degree objects that areacceptable to the target modeler such that when these lower degreeobjects are combined, they approximate the original high degree object.However, such a conversion presents its own set of difficulties. Inparticular, there may be both computational efficiency difficulties andpossibly difficulties maintaining certain tolerances in the targetmodeler. Moreover, there is very likely to be inaccuracies in theconversion process. For instance, the speed of computation in downstreamprocesses within such a target modeler after such a conversion of ageometric object may grow greater than linearly with the degree of theobject representation being converted. For example, a geometric objecthaving representation with degree 50 may, when converted to, e.g., aplurality of 25 degree approximations subsequently require, e.g., fourto five times the computational time that a geometric object representedby a polynomial representation of 25 degrees.

Accordingly, it would be very desirable to have a CAD system thatincludes one or more geometric design techniques for allowing CADdesigners/users to more easily, efficiently and precisely designgeometric objects. Further, it would be desirable to have such a systemand/or computational techniques for graphically displaying geometricobjects, wherein there is greater user control over the defining and/ordeforming of computational geometric objects, and in particular, moreintuitive global control over the shape or contour of computationallydesigned geometric objects. Additionally, it would be desirable toprovide surfaces that smoothly blend into one another, wherein thesurface representations have a low polynomial degree.

Moreover, one important quality of a surface is its smoothness. However,it is nontrivial to quantitatively measure what is meant by“smoothness”. For example, a know technique to measure the smoothness ata join of two surfaces is to use the continuity of the derivatives ofthe surface generating functions for those two surfaces. However, thismeasure of smoothness does not always work for parametric functions usedin design. For instance, sometimes two functions can form a smooth curveor surface while their-derivatives do not agree at a joint between thecurves or surfaces. On the other hand, there could be a sharp cusp evenif the derivatives are the same. In order to describe “smoothness”,certain terms are used in the art. In particular, various continuitycriteria are used to describe smoothness. More precisely, let there betwo parametric curves f(u) and g(v), where u and v are in the intervals[a,b] and [m,n], respectively. For describing the smoothness of a joinof these curves, consider the “right-end” of curve f(u), i.e., f(b), andthe “left-end” of curve g(v), i.e., g(m). If f(b)=g(m), we say curves fand g are C⁰ continuous at f(b)=g(m). If for all i<k, the i^(th)derivatives at f(b) and g(m) are equal, we shall say that the curves areC^(k) continuous at point f(b)=g(m).

Curvature is another measure that can be used to evaluate the smoothnessbetween curves (and eventually surfaces). In particular, for aparametric function ƒ the curvature is defined byk=|ƒ′(t)×ƒ″(t)|/|ƒ′(t)|³  (1)

If the curvatures of curves at a join thereof are equal, we will saythey are curvature continuous at the join. Intuitively, two curves arecurvature continuous if the turning rate is the same at the join;however, the second derivatives of these curves may not be the same atthe join. In other words, curvature continuity is not always C²continuity with parametric functions. For example, the following curveconsists of two parabolic segments:f(u)=(u,−u ²) and g(v)=(v,v ²),where the left curve f and the right curve g have domains [−1,0] and[0,1], respectively.

Consider the continuity of the two curves at the origin where they join,i.e.,f′(u)(1,−2u)f″(u)=(0,−2)g′(v)=(1,2v)g″(v)=(0,2)Since f′(0)=g′(0)=(1, 0), these two curves are C′ continuous at theorigin. However, since f″(0)=(0, −2) is not equal to g″(0)=(0, 2), theyare not C² continuous at the origin. In fact, f″(u) points to the southwhile g″(u) points to north, and both are constant. Therefore, when amoving point crosses the origin from one curve to the other, the secondderivative changes its direction abruptly. However, the curvature off(u) is k_(f)=2/(1+4u²)^(1.5) and the curvature of g(v) isk_(g)=2/(1+4v²)^(1.5). Accordingly, the two curve segments are curvaturecontinuous at the join. Thus, we see that two curve segments may be C¹and curvature continuous, but not C² continuous.

Consider yet another simple example. Letf(u)=A+u(B−A)g(v)=B+v(C−B)where A, B and C are three collinear points as shown in FIG. 1.

The functions f(u) and g(v) are obviously C⁰ continuous at the join B.Is it C¹ continuous? We have:f′(u)=B−Ag′(v)=C−B

Clearly, f′(u)=B−A is not equal to g′(v)=C−B, for most values of A, B,and C. Consequently such line segments are, in general, not C¹continuous at B.

This poses a problem for parametric forms that is not seen in explicitforms where the derivative is a scalar—the slope. It is a result of thederivative being a vector with both magnitude and direction. If wereplace the direction vectors B−A and C−B with unit-length vectors andchange the domain of parameters u and v, this problem will disappear.That is, the above equations are re-parameterized to the following:F(u)=A+u(B−A/|B−A|G(v)=B+v(C−B)/|C−B|where u∈[0,|B−A|] and v∈[0,|C−B|]. Now, since we have F′(u)=G′(v) equalto the unit-length vector in the direction of A to C, the line segmentsare C¹ continuous. Thus, re-parameterizing the curve segments overcomesthe problem.

Many C¹ continuous curves are curvature continuous, but they are not C²continuous at their join(s); some of them may not even be twicedifferentiable. However, these curves can appear smooth at the join(s),and having a smooth join is an important feature of computationaldesign.

Therefore, we introduce the notion of Geometric Continuity defined inthe following:

Two curve segments are said to be G^(k), equivalently, G^(k) geometriccontinuous, and G^(k) continuous (and have G^(k) Geometry Continuity) attheir join if and only if there exist two parameterizations, one foreach curve segment, such that all i^(th) derivatives, i≤k, agree at thejoin. Reference: Hoschek and Lasser.

The idea of G^(n) geometric continuity (n≥1) may be extended to surfacesas follows: two surfaces are G^(n) geometric continuous at a jointhereof if and only if for each point of the join of the two surfaces,each curve on the surfaces that crosses the join at the point is G^(n)geometric continuous. Note that in the disclosure hereinbelow a surface(S) is described as being a G^(n) surface or G^(n) continuous when thereis G^(n) geometric continuity for each joined pair of surfaces in S.Moreover, when there is no ambiguity, the surface S may be simplyreferred to as a “G^(n) surface”.

G² geometric continuity is highly desirable for creating smoothness indesign because it makes surfaces with smooth reflections and improvesfluid flow among other advantages.

Therefore it would be desirable to have a computationally effective andefficient technique for describing “smoothness”, and in particular,using such a technique for generating designs of objects whose compositesurfaces are visually smoother than typically provided by the prior art.

Definitions

This section provides some of the fundamental definitions that are usedin describing the computational features disclosed herein. Thesedefinitions are also illustrated in FIGS. 15 and 16.

A “parametric geometric object” S is a geometric object that is theimage of a function f, wherein the domain of f is in a geometric spaceembedded within a coordinate system (denoted the “parameter space”) andthe range of f is in another geometric space (denoted the “objectspace”). Typically, the inverse or pre-image, f⁻¹, of a geometric objectsuch as S will be a geometrically simpler object than its image inobject space. For example, the pre-image of a curve 170 in object spacemay be a simple line segment 172, L, in parameter space. Thus if Sdenotes the curve in object space, then notationally f and S aresometimes identified such that for u∈L, a corresponding point in thecurve S is denoted S(u). Similarly, the pre-image of an undulatingsurface 204 (FIG. 16) in object space may be a simple bounded plane 180in parameter space. Thus, if S denotes the undulating surface 204, thenfor (u,v)∈f⁻¹(S), S(u,v) denotes a corresponding point on the undulatingsurface 204.

A “profile” 200 (FIG. 16) is a geometric object, such as a curve inobject space, through which an associated object space geometricallymodeled object (e.g. surface 204) must pass. That is, such profiles 200are used to generate the geometrically modeled object. Thus, profilesprovide a common and natural way for artists and designers togeometrically design objects, in that such a designer may think in termsof the feature or profile curves when defining the characteristic shapeof a geometric object (surface) being designed. For example, profilecurves on a surface may substantially define the geometry of a resultingderived geometric object; e.g., its continuity, curvature, shape,boundaries, kinks, etc. Note, that for many design applications,profiles are typically continuous and differentiable. However, suchconstraints are not necessary. For example, a profile may, in additionto supplying a general shape or trend of the geometric object passingtherethrough, also provide a texture to the surface of the geometricobject. Thus, if a profile is a fractal or fractal-like, the fractalcontours may be in some measure imparted to the surface of the derivedgeometric object adjacent the profile. Further note that it is withinthe scope of the present disclosure to utilize profiles that are ofhigher dimension (≥2). Thus, a profile may also be a surface or a solid.Accordingly, if a profile is a surface, then a solid having locally(i.e., adjacent to the profile) at least some of the geometriccharacteristics of the profile may be derived.

Moreover, profiles (and/or segments thereof) may have variouscomputational representations such as linear (e.g., hyperplanes),elliptic, NURBS, or Bezier. Note, however, that regardless of thecomputational representation, a method (such as interpolation) fordeforming or reshaping each profile is preferable. More particularly, itmay be preferable that such a method results in the profile satisfyingcertain geometric constraints such as passing through (or substantiallyso) one or more predetermined points, being continuous, beingdifferentiable, having a minimal curvature, etc. Further, note that sucha deformation method may also include the ability to decompose a profileinto subprofiles, wherein the common boundary (e.g., a point) betweenthe subprofiles may be “slidable” along the extent of the originalprofile.

A “marker” 208 (FIG. 16) is a point on a profile that can be moved tochange the shape of the profile 200 in a region about the marker. Amarker also designates a position on a profile where the shape of ageometric object having the profile thereon can be deformed.

A “profile handle” 212 (FIG. 16) is a geometric object tangent to theprofile 200. Such a profile handle may control the shape of the profilelocally by modifying the slope (derivative) of the profile at the marker208. Alternatively, for non-differentiable profiles, a profile handlemay be used to control the general shape of the profile by indicating atrend direction and magnitude of the corresponding profile. For example,if the profile is a fractal or other nondifferentiable geometric object,then a profile handle may, for example, provide a range within theobject space to which the profile must be confined; i.e., the range maybe of a tubular configuration wherein the profile is confined to theinterior of the tubular configuration. Note that the profile handle 212affects the fullness of the profile 200 (e.g., the degree of convexitydeviating from a straight line between markers on the profile) bychanging the length of the profile handle.

An “isocline boundary” 220 is the boundary curve opposite the profile200 on the isocline ribbon 216. In one embodiment, at each point on theprofile 200 there is a paired corresponding point on the isoclineboundary 200, wherein each such pair of points defines a vector 224(denoted a “picket”) that is typically transverse to a tangent vector atthe point on the profile. More particularly, for a parameterizedprofile, the isocline boundary 220 can be viewed as a collection ofpickets at all possible parameter values for the profile 200.

An “isocline ribbon” (or simply isocline) is a geometric object, such asa surface 216, which defines the slope of the geometric object (e.g.,surface) 204 (more generally a geometric object 204) at the profile 200.Equivalently, the isocline ribbon may be considered as therepresentation of a geometric object delimited by the profile 200, theisocline handles 218 a and 218 b (discussed hereinbelow), and theisocline boundary 220. In other words, the geometric object 204 must“heel” to the isocline ribbon 216 along the profile 200. Said anotherway, in one embodiment, the geometric object 204 must be continuous atthe isocline 216 and also be continuously differentiable across theprofile 200. In an alternative embodiment, the geometric object 204 maybe constrained by the isocline 216 so that the object 204 lies within aparticular geometric range in a similar manner as discussed above in thedescription of the term “profile.” Note that there may be two isoclineribbons 216 associated with each profile 200. In particular, for aprofile that is a boundary for two abutting surfaces (e.g. two abuttingsurfaces 204), there can be an isocline ribbon along the profile foreach of the two surfaces. Thus we may speak of a right and a left handisocline ribbon.

An “isocilne handle” 228 is a geometric object (e.g., a vector) forcontrolling the shape of the isocline ribbon 216 at the marker 208,wherein the profile handle and isocline handle at the marker may definea plane tangent to the surface 204. Hence the isocline handle is used todetermine the shape of the surface 204 (or other underlying geometricobject) about the marker. In particular, an isocline handle 228 is auser manipulatable picket 224. If all the profile handles 212 andisocline handles 228 (e.g., for two or more abutting surfaces) arecoplanar at a marker 208, then the surface 204 will be smooth at themarker (assuming the surface is continuously differentiable), otherwisethe surface may have a crease or dart. Note that by pulling one of thehandles (either isocline or profile) out of the plane of the otherhandles at a marker, one may intentionally generate a crease in thesurface 204 along the profile 200.

The part of the profile 200 between two markers 208 is denoted a“profile segment” 232. Similarly, the part of the isocline ribbon 216between two isocline handles 228 is denoted a ribbon segment 240.

A “boundary segment” 244 denotes the part of the boundary 220 betweentwo isocline handles 228.

The vector 246 that is the derivative tangent to the isocline boundary220 at an isocline handle 228 is denoted a “ribbon tangent.” Note thatmodifications of ribbon tangents can also be used by the computationalsystem of the present disclosure to control and/or modify the shape ofan underlying geometric object such as surface 204.

Isocline handles 228 may be generalized to also specify curvature of thesurface 204. That is, instead of straight vectors as isocline handles,the handles may be curved and denoted as “isocline ribs” 248. Thus, suchribs may facilitate preserving curvature continuity between surfaceshaving associated isocline ribbons along a common profile boundary,wherein the isocline ribbons are composed of isocline ribs. Accordingly,the curvature of such surfaces will match the curvature of theircorresponding isocline ribs, in much the same way as they match intangency.

A “developable surface” is a surface that can be conceptually rolled outflat without tearing or kinking. It is a special case of a “ruledsurface,” this latter surface being defined by being able to lay a ruler(i.e., straight edge) at any point on the surface and find anorientation so that the ruler touches the surface along the entirety ofthe ruler. For a developable surface, the surface perpendiculars are allequal in direction along the ruling.

“Label surfaces” denote special 2-dimensional (developable or nearlydevelopable) surfaces wherein a label may be applied on, e.g., acontainer. Label surfaces allow application of a decal without tearingor creasing. These surfaces are highly constrained and are not typicallydeformed by the geometric modification of an isocline ribbon 216.

A “trim profile” is a geometric object (curve) that is a profile fortrimming another geometric object (e.g., a surface). The trim profilemay have a single corresponding isocline ribbon 216 since if the surfaceto be trimmed is a label surface, it will not be modified and,accordingly, no isocline ribbon can be used to change its shape.

A trim profile (curve) can be used to delimit any surface, not just alabel surface. A surface, S, that is blended along a trim profile withone or more other surfaces is called an “overbuilt surface” when thesurface S overhangs the trim profile. For example, in FIG. 11, surface130 is an overbuilt surface, wherein the portion of the surface outsideof the area 134 is typically not shown to the designer once it has beentrimmed away.

A convex combination of arguments F_(i) is a summation

$\sum\limits_{i}{c_{i}F_{i}}$where the c_(i) are scalar coefficients and scalar multiplication iswell-defined for the F_(i) (e.g., F_(i) being vectors, functions, ordifferential operators), and where c_(i)≥0 and

${\sum\limits_{i}c_{i}} = 1.$If the F_(i) are points in space, for instance, then the set of allpossible such combinations yields the convex hull of the points F_(i),as one skilled in the art will understand.

A “forward evaluation” is a geometric object evaluation technique,wherein in order to generate a set of sample values from a function,f(x), argument values for x are incremented and f is subsequentlyevaluated. This type of evaluation is usually fast and efficient, butdoes not give function values at chosen positions between theincrements.

An “implicit function” is one written in the form f(x)=0. χ∈

^(N) When a parametric curve or surface is converted to an implicitform, the conversion is called “implicitization.” Hence f(t)=(sin(t),cos(t)) in parametric form may be implicitized by f(x,y)=x²+y²−1=0. Bothforms describe a circle.

Dividing a vector by its length “normalizes” it. The normalized vectorthen has unit length. A vector function may be divided by its gradient,which will approximate unit length, as one skilled in the art willunderstand.

Given a function defined by a

$\sum\limits_{i}{{p_{i}(t)}{F_{i}(t)}}$where p_(i)(t) are weighting functions, if

${\sum\limits_{i}{p_{i}(t)}} = 1$for all values of t, then the p_(i) are said to form a “partition ofunity.”

“G1” continuity denotes herein a geometric continuity condition whereindirection vectors along a continuous parametric path on a parametricallydefined geometric object are continuous, e.g., tangent vector magnitudesare not considered.

SUMMARY OF THE INVENTION

The present disclosure presents a computational geometric design systemthat is capable of sufficiently efficient computations so as to allowreal-time deformations to objects such as surfaces while a user issupplying the object modifying input. Thus, the present disclosurepresents a paradigm shift away from typical CAD systems since, in atypical CAD system the user must supply input for changing or modifyinga geometric object and subsequently explicitly request processing of theinput to commence. Thus, in such prior art CAD systems, the userreceives feedback about his/her design at discrete user requested times.Instead, with the computational system of the present disclosure,updates may be processed in real-time immediately upon input receiptwithout the user explicitly indicating that update processing is to beperformed.

Given the enhanced computational efficiency of the present disclosure, auser can more efficiently perform iterative approximations to ageometric object being designed. The user may speedily design withoutthe need to precisely calculate design geometric characteristics forportions of the object where such precision may not be necessary. Thatis, the user can be less concerned about getting it “right the firsttime” since the ease of modification and speed of computingmodifications allows the user to more rapidly approximate and/orprototype a geometric object. Thus, the present disclosure providessubstantial efficiency benefits in that for many geometrically designedobjects (including machined parts), substantial portions of such objectsmay be satisfactorily designed with a wide range of geometriccharacteristics.

A CAD system according to the present disclosure enables novel designtechniques by providing a novel computational technique for blendingbetween two parametric geometric objects such as surfaces. In oneembodiment of the present disclosure, this novel blending techniqueblends between two parametric surfaces S₁(u,v) and S₂(u,v), wherein eachsurface has, e.g., the unit square as its parameter space. Assuming eachsurface S₁ and S₂ has a respective blending function B₁(u,v) and B₂(u,v)such that each of the blending functions has, for example, (0,1) as itsrange for u and v (as well as satisfying other properties givenhereinbelow), a new surface, S, may be defined by the following formula:S(u,v)=S ₁(u,v)·B ₁(u,v)+S ₂(u,v)·B ₂(u,v)  (1)Note that the blending functions B₁ and B₂ are typically chosen so thatthe resulting blended surface S is the same as S₁ on a boundary with S₁,and the same as S₂ on a boundary with S₂. This is achieved by devisingB₁ and B₂ so that B₁=1 and B₂=0 on the boundary with S₁ while havingB₂=1 and B₁=0 on the boundary with S₂.

In a more general embodiment, an embodiment of the present disclosuremay be used for blending between a plurality of geometric objects,S_(i), i=1, . . . , N, wherein each of the geometric objects isparameterized by a corresponding function f_(s) _(i) whose domainincludes a parameter space PS common to all f_(s) _(i) . Thus, for theresulting blended surface S, substantially every one of its points,S(q), for q in PS is determined using a weighted sum of points obtainedfrom the points f_(s) _(i) (q). Moreover, since it is desirable to blendS to a boundary portion P_(i) of each S_(i), when interpreting S as afunction from PS to the common geometric space GS having the geometricobjects S_(i), then S(f_(s) _(i) ⁻¹(P_(i)))⊆P_(i). Additionally, S maybe also continuous at each f_(s) _(i) ⁻¹(P_(i)).

Note that Formula (1) above is representative of various formulas forgenerating a blended surface (more generally, geometric object) S, otherembodiments of such formulas are provided in the Detailed Descriptionhereinbelow. Further note that such formulas may be generalized whereinthe parameter space coordinates (u,v) of Formula (1) above can bereplaced by representations of other parameter space coordinates such astriples (u,v,w) or merely a single coordinate u. Moreover, the blendingfunctions B₁ and B₂ may also be defined for such other parameter spaces.Additionally, it is worthwhile to note that such blending functions B₁and B₂ may be considered as weights of a weighted sum of points selectedfrom the surfaces (more generally, geometric objects) S₁ and S₂.Further, this weighted sum notion may also be extended in various ways.For example, referring to the more general embodiment wherein aplurality of geometric objects S_(i), i=1, . . . , N are provided, acorresponding weight/blending function B_(i) may be provided for eachvalue of i so that the following variation of Formula (1) is obtained:

${S(q)} = {\sum\limits_{i = 1}^{N}{{B_{i}(q)}{S_{i}(q)}}}$for points q in a common parameter space for the S_(i).

In another aspect of the present disclosure, it is within its scope toalso generate blended geometric objects S, wherein at least some of theS_(i) geometric objects used to generate S are of a higher dimensionthan 2. For example, if S₁ and S₂ are parameterized solids, then S maybe generated as a solid blended from S₁ and S₂ using another variationof Formula (1), as one skilled in the art will understand. Thus, S mayextend between S₁ and S₂ so that a surface P₁ of S₁ and a surface P₂ ofS₂ are also boundaries of S, and S is represented as a weighted sum ofpoints of S₁ and S₂ similar to Formula (1).

In yet another aspect of the present disclosure, one or more of theparametric geometric objects S_(i) of Formula (1) (or variations thereofsuch as Formula (2), (4), (5), (5.02), (5.03), etc. provided in theDetailed Description hereinbelow) may have one of: a Bezier, NURBS, orsome other multivariant parametric computational representation insteadof, e.g., isocline ribbons as illustratively used in the descriptionherein. Moreover, as one skilled in the art will also understand, it iswithin the scope of the present disclosure that the underlying geometricobjects that define the S_(i)'s (e.g., for the S_(i) being isoclineribbons, such underlying geometry being markers, profiles, isoclinehandles and profile handles) may be different for a differentcomputational representation. For example, in a Bezier or NURBSrepresentation of an S_(i) “control points” and/or geometric entitiesderived therefrom, may be used to change a shape of the S_(i) andtherefore change the shape of the resulting geometric object S derivedtherefrom.

In another aspect of the present disclosure, when a blended surface S isgenerated from one or more isocline ribbons S₁, . . . , S_(N), N≥2, thesurface S can be deformed by changing geometric characteristics of theisocline ribbons S_(i). In particular, by changing the shape of one ofthe isocline ribbon boundaries for some S_(i), the points S_(i)(u,v)change and accordingly, the blended surface S changes since it is aweighted sum of such points. In particular, rates of change of geometriccharacteristics of S (such as curvature, tangent vectors, and/or tangentplanes) may be determined by the shape of the isocline ribbons S_(i).More particularly, assuming a substantially linear parameterizationalong each isocline picket, the greater the relative magnitude of suchpickets for a particular isocline ribbon, the more the shape of S willbe skewed in the direction(s) of such pickets. Moreover, as thedirection of such pickets changes, the curvature of S changes. That is,since the weighted sums, such as Formula (1), cause S to always heel tothe surfaces S_(i), the shape of S will change so that S heels to theisocline ribbon(s) S_(i) having pickets whose directions have changed.Thus, the shape of the blended surface S may be changed by any userinteraction technique that: (a) changes one or more geometriccharacteristics of one or more of the S_(i), wherein such changes mayinclude: changing a shape of S_(i) (wherein shape denotes a plurality ofgeometric characteristics such as continuity, differentiability,curvature, and higher order continuity), (b) changes a parameterizationof an S_(i).

Also note that such user interaction techniques for deforming a blendedsurface may also be used with higher dimensional geometric objects. Forexample, if the S_(i) geometric objects are solids rather than surfaces,then a resulting blended solid, S may be deformed by changing a shape ofone or more of the solids S_(i) used in determining S.

It is also within the scope of the present disclosure that the geometricobjects S_(i) used to generate a blended geometric object S may be suchthat the S_(i)'s can be modified indirectly via other geometric objectsfrom which the S_(i)'s may be themselves generated. For example, if S isa surface blended from isocline ribbons S₁ and S₂ (having correspondingprofiles P₁ and P₂, respectively), and the ribbon S₁ is interpolatedfrom the profile handle, the isocline handle, and the ribbon tangent atthe end points of P₁, then the present disclosure provides userinteraction techniques for modifying such handles and/or ribbon tangentsfor thereby modifying the blended surface S. Moreover, in one userinterface technique, only the handles may be displayed, wherein suchhandles are displayed as connected to the blended surface S. Thus, bychanging such handles, the blended surface changes. Note that such userinteraction techniques may be responsive in real time to user changes tosuch handles and/or ribbon tangents. Thus, a user's design intent may beimmediately displayed while the user is inputting such changes.Accordingly, using the present disclosure, user interactions in thedesign process may become closer to the techniques in used inconstructing actual geometric models rather than prior art CAD userinteraction techniques.

It is another aspect of the present disclosure that various geometricconstraint criteria are capable of being applied to geometric objectsgenerated according to the present disclosure. In particular, featuresand/or subgeometry of a geometric object O₀ are capable of beingconstrained to lie within another geometric object, O₁, so that as O₁ isdeformed, the features and/or subgeometry of O₀ deform correspondingly,and thereby cause O₀ to deform accordingly. For example, the presentdisclosure allows an object space point p to be defined (i.e.,parameterized) so that it must remain in/on a given geometric object O₁,where O₁ may be a curve, surface, volume or solid. Thus, as O₁ isdeformed, O₀ also deforms. Moreover, instead of a point p, othergeometric subobjects may also be similarly constrained, such as curves,surfaces or solids. Additionally, features of a geometric object O₀ suchas control points, handles (of various types, e.g., profile andisocline), normals, twist vectors, etc. may also be similarlyconstrained by the present disclosure so that as O₁ is deformed, O₀ iscaused to also deform. For instance, using the geometric objectinterpolation techniques provided by the present disclosure, e.g.,Formula (1) and variations thereof, the geometric object O₀ can beefficiently regenerated (e.g., reinterpolated) substantially inreal-time when constrained features and/or subgeometries of O₀ arecorrespondingly deformed with a deformation of O₁. More particularly,this aspect of the present disclosure provides for the combining ofvarious geometric objects hierarchically so that geometric deformationcontrol of a parent object causes corresponding geometric changes independent child geometric objects. For example, when a surface patchrepresents fine scale detail of a larger surface, it may be advantageousto attach the fine detail surface patch to the larger surface to therebygive a user automatic control over the shape of the fine detail surfacepatch by controlling the shape of the larger surface. Moreover, similarhierarchical control can be provided with other geometric objects oftypes such as curves, points and three-dimensional deformation spaces.

Note that such hierarchical control may be also used with a persistentdeformation space wherein it is desirable for a geometric object in thisspace to be repeatedly deformed and restored to its originalnon-deformed state. Note that this is difficult to do in real-time byrepeatedly applying a one-time deformation. Accordingly, by utilizingsuch hierarchical control of the present disclosure, a geometric objectembedded in such a three-dimensional deformation space and/or thecontrol structures of the geometric object embedded therein provides forthe deformation of the geometric object when the three-dimensionaldeformation space is deformed. Further, if one or more such deformationspaces are, in turn, made dependent upon a simpler geometry such as asurface or curve, then substantial control over the shape of thegeometric object, however complex, can be provided by manipulating theshape of the simpler geometry.

A novel aspect of the present disclosure is an N-sided surfacegeneration technique which is applicable for efficiently and accuratelyconverting surfaces of high polynomial degree into a collection of lowerdegree surfaces. In particular, the N-sided surface generation techniquedisclosed herein subdivides parameter space objects (e.g., polygons) ofseven or more sides into a collection of subpolygons, wherein eachsubpolygon has a reduced number of sides. More particularly, eachsubpolygon has 3 or 4 sides.

In one embodiment, for a parameter space object having N sides (N>7),the novel technique disclosed herein determines a point in a centralportion of the parameter space object, and an edge to each vertex of theobject is introduced for subdividing the object. Such subdivisioncreates N triangular patches. As an illustrative example, FIG. 2 showsan octagon 10 in parametric space that is subdivided into eight 3-sidedareas 12 a-12 h, wherein for each of these areas 12 a-12 h of theparameterization domain, the N-sided surface generation techniquedisclosed herein provides a corresponding one of the mappings 14 a-14 hto a corresponding patch 16 a-16 h in object space (e.g., area 12 c mapsto patch 16 c via mapping 14 c). Moreover, the resulting object spacepatches 16 a-16 h are advantageously watertight, i.e., there are no gapsbetween the images of the parameter space subdivisions where:

-   -   these subdivisions have a common edge, and    -   such gaps would allow water to leak through if the surface were        manufactured out of a water tight material.        Such patches, resulting from the application of the present        N-sided surface generation technique to parameter space        subdivisions having a common side, form a composite surface        having a smooth change in surface tangency across the (object        space) boundaries of these patches. In particular, and        importantly, composite surface tangents along the image of a        common boundary between parameter space subdivisions are        identical regardless of how such tangents are determined in the        composite surface.

The novel N-sided surface generation technique disclosed hereingenerates analytic representations of N-sided object space surfaces(each surface also referred to herein as an NSS), wherein the polynomialdegree of such an NSS is reduced over what is provided in the prior art.Such reduction in the polynomial degree of an NSS results insubstantially greater computational efficiency in the generating ofsurfaces, and in performing geometric modeling therewith. Additionally,the present N-sided surface generation technique can be used inconverting object space surfaces of high polynomial degree tocorresponding surfaces of lower degree, wherein each of the convertedsurfaces is effectively a copy of its corresponding surface of highpolynomial degree in that the discrepancies between the two surfaces canbe made insignificant. Accordingly, when the present the N-sided surfacegeneration technique is embodied as a converter for converting surfacesof high polynomial degree to a composite of surfaces of lower polynomialdegree, interoperability between various geometric modeling systems isfacilitated. In particular, in a geometric modeling system thatgenerates at least some surfaces of high polynomial degree, suchsurfaces can be converted into a corresponding collection of surfaceshaving lower polynomial degrees, and accordingly, data for the resultingcomposite surface obtained from the collection can be appropriately usedby a commercial geometric modeler that would be unable to appropriatelyuse the corresponding surface of high polynomial degree. Moreparticularly, since the generated N-sided surfaces of low polynomialdegree can be further converted into, e.g., trimmed Bezier surfacerepresentations and/or trimmed non-uniform b-spline surface (NURBS)representations, the present surface generation technique effectivelyallows for the use of surfaces having excessively high polynomialdegrees to be acceptable to commercial surface generation computationalsystems such as those provided by Alibre™, Solidworks™, and SpatialTechnologies™.

The conversion process provided by the present N-sided surfacegeneration technique may be used to effectively import into a commercialgeometric modeler (expecting all geometric representations to have a lowpolynomial degree, e.g., less than or equal to 25) an excessively highpolynomial degree surface. In particular, surfaces having a parametricpatch with a large number of sides, e.g., greater than seven sides, maybe effectively imported into such a geometric modeler by converting suchexcessively high degree surface patches into a collection of lowerdegree patches that match the high degree surface as closely as desired.Thus, such a conversion is more effective than prior art conversiontechniques.

In one embodiment of the novel surface generation technique disclosedherein, this technique approximates each N-sided patch, for N>7, by acollection of 3 and/or 4-sided patches, wherein the computationalreliability and modeling accuracy can be maintained. In particular, themodeling accuracy provided by the present conversion is such that theerror between the original N-sided patch of N>7, and the collection of 3and/or 4-sided patches can be made as small as the computercomputational tolerance will allow. Moreover, when the presentlydisclosed technique is applied to N-sided patches where N=3 to 7, theconversion is exact. In particular, note that 4-sided patches have anexact conversion, which is important because such patches are ofreasonably low degree, and many useful engineering surfaces such asplanar, extrusion and rotational surfaces can be exactly reproduced with4-sided patches.

In one embodiment, novel surface generation technique disclosed hereinperforms the following steps:

Step 1: Divide N-sided patches into 3-sided and/or 4-sided patches,where N>4;

Step 2: Convert the 3 or 4 sided patches (with or without correspondingsurface tangent information), using conversion matrices, into Bezierpatches; and

Step 3: Convert Bezier patches into B-spline patches.

Moreover, the presently disclosed surface generation technique providesadvances in such computationally difficult geometric modeling issues aswater tightness, iso-parameter behavior and tangency due to theincreased exactness of surface conversions afforded by the presentlydisclosed surface generation technique.

Furthermore, the present disclosure provides techniques for G²continuity (also known as “G2 geometry continuity”) between suchpatches, which produce surfaces with improved visual and functionalcharacteristics.

The computational features disclosed herein can be applied to industriessuch as, but not limited to, internet advertising, animation, automotivestyling, CAD/CAM, product design, and computer gaming models, etc.

Other features and benefits of the present disclosure are provided inthe Detailed Description and the drawings provided herewith.

BRIEF DESCRIPTION OF THE DRAWING

The file of this patent contains at least one drawing executed in color.Copies of this patent with color drawings will be provided by the Patentand Trademark Office upon request and payment of the necessary fee.

FIG. 1 shows a surface 62 generated according to the present disclosure,wherein the surface interpolates between the surfaces 30 and 34, andalso passes through the curves 54, 58 and 60 at predetermined directionsaccording to the isocline ribbons 61 and 63.

FIG. 2 shows a further modification of the surfaces of FIG. 1, whereinthe surface 30 has a circular disk 66 blended thereto according to amethod of the present disclosure.

FIG. 3 shows a blended surface 62 a generated according to the presentdisclosure between the surfaces 30 and 34, wherein the surface 62 apasses through the curves 54 and 58 and wherein the blending isperformed according to a novel surface generation formula providedherein (Formula (1)).

FIG. 4 illustrates a correspondence between geometric entities inparameter space and geometric entities in object space, wherein lines 78a and 78 b of parameter space have object space images of curve 54 and58, respectively, and additionally, parameter space line 86 has as anobject space image curve 80.

FIG. 5 provides a graphical representation of two blending functions, B₁and B₂, utilized in some embodiments of the present disclosure.

FIGS. 6A-6D show graphs of additional blending functions that may beused with the present disclosure.

FIG. 7 provides a further illustration of the correspondences betweengeometric entities in parameter space and object space.

FIG. 8 shows an elliptic region 100 that is blended into a cylinder 108according to the present disclosure, wherein the closed curve 110delimits the elliptic region from the deformed portion of the cylinder108 that blends to the closed curve.

FIG. 9 shows a simple boss 112 created on a cylinder 116 according tothe method of the present disclosure.

FIG. 10 shows a composite curve 120 (as defined hereinbelow) thatincludes two crossing subcurves 124 and 128.

FIG. 11 shows a surface 130 from which a label surface 134 is trimmed.

FIG. 12 illustrates one computational technique for determining adistance-like measurement from a point p that is interior to the polygonhaving vertices v₁, v₂, v₃, v₄, and v₅, wherein the distance-likemeasurement to each side of this polygon is determined using acorresponding apex 150 provided by a stellating process.

FIG. 13 shows two boundary curves 156 a and 156 b in parameter space(i.e., the unit square), wherein area patches 168 therebetween arecapable of being themselves parameterized by coordinates (s,t) where svaries linearly with the distance between two corresponding points on apair of opposing subcurves 160 a and 160 b, and t determines acorresponding point on each of the subcurves of the pair 160 a and 160b.

FIG. 14 illustrates a region that has sides and ribbons defined by threesurfaces S₁, S₂ and S₃, wherein the present disclosure shows a surfacepatch for the region 300 using Formula (5) provided hereinbelow.

FIGS. 15 and 16 illustrate both general computational geometry concepts,as well as novel concepts that are fundamental to the presentdisclosure. Note that these figures are used to illustrate the termsdefined in the Definitions Section hereinabove.

FIG. 17 shows a block diagram of the typical flow of design constructionoperations performed by a user of a computational system according tothe present disclosure in designing a geometric object.

FIG. 18 shows three profile curves 404, 408 and 412 meeting at a profilemarker 420, wherein the surfaces 416 and 418 join smoothly at the marker420 due to the isocline handles (for the marker 420) being in a commonplane 460.

FIG. 19 shows profile curves x and y that define a surface 480 whichforms a fillet between surfaces 484 and 486. Typically, profiles x and yare defined using distances 488 and 490 from the intersection curve 482of surfaces 484 and 486.

FIG. 20 illustrates one embodiment for computing a blended surface fromisocline ribbons 508 and 516 according to the present disclosure.

FIGS. 21A-21C illustrate a procedure for creating a hole 600 accordingto the present disclosure.

FIG. 22 shows a blended surface 710 according to the present disclosure,wherein the blended surface extends between a degenerate profile (point)714 and the circular end 718 of a cylinder 722.

FIG. 23 shows a blended surface 750 according to the present disclosurethat extends between the degenerate profile (point) 754 and the planardisk 758 having a circular curve 760 therein.

FIG. 24 illustrates the results of a blending technique of the presentdisclosure for blending a surface between semi-circular ribbons 784 aand 784 b, wherein the resulting surface 786 is blended between thesetwo ribbons.

FIG. 25 shows a blended surface 808 according to the present disclosurewhose points p(u,v) are determined using a “forward algorithm”, whereinpoints in parameter space 158 are themselves parameterized according topoints in an additional parameter space 828, and wherein the points 830of the additional parameter space are used to efficiently determine thedistance-like measurements to the pre-images (in parameter space 158) ofthe profiles 812 and 816 (in object space).

FIG. 26 is a flowchart showing the steps for computing an interpolatingcurve according to the present disclosure using a one-dimensionalembodiment of the computational techniques novel to the presentdisclosure.

FIG. 27 shows a flowchart of the steps performed when constructing anapproximation to an isocline boundary of an isocline ribbon, wherein theboundary is opposite the profile for the isocline ribbon.

FIGS. 28A and 28B show a flowchart for a program that constructs a moreprecise isocline ribbon boundary than the approximation resulting fromFIG. 27.

FIGS. 29A-29C illustrate a flowchart for modifying one or moresubsurfaces S_(i) of a composite surface S₀ by changing a geometriccharacteristic of an isocline handle and/or a ribbon tangent for amarker on one or more profile curves defining the boundaries for thesubsurfaces S_(i).

FIGS. 30A and 30B provide a flowchart of a program invoked by theflowchart of FIG. 29 for deforming subsurfaces S_(i) in real time as auser modifies an isocline handle and/or ribbon tangent.

FIG. 31 is a flowchart of the high level steps performed by a userinteracting with an embodiment of the present disclosure for changingthe shape of a surface.

FIG. 32 pictorially illustrates examples of values for parameters usedin the flowchart of FIG. 26 for computing an interpolating curve C(u).

FIG. 33 shows four profile curves P₁₁, P₁₂, P₂₁ and P₂₂ wherein it isdesired to generate a surface, bounded by these profiles and wherein thesurface is defined by these four profiles (and their associated isoclineribbons).

FIGS. 34 and 35 illustrate the intermediary surfaces generated duringthe performance of one method for creating a 4-sided patch (FIG. 36)from two 2-sided blends using the four profile curves of FIG. 33. Thatis, a blended surface S₁ (FIG. 34) is generated using the isoclineribbons R₁₁ and R₁₂ (for the profiles P₁₁ and P₁₂, respectively), and ablended surface S₂ (FIG. 35) is generated using the isocline ribbons R₂₁and R₂₂ (for the profiles P₂₁ and P₂₂, respectively).

FIG. 36 shows a resulting blended surface S derived from S₁ (shown inFIG. 34), and S₂ (shown in FIG. 35), wherein S is generated according toFormula (11) provided in Section 5 of the detailed descriptionhereinbelow.

FIG. 37 shows the geometric objects used in an embodiment of the presentdisclosure for generating a surface S from two surfaces S_(L) and S_(R).In particular, this figure introduces the notational conventionssubsequently used in FIGS. 38 through 41.

FIG. 38 illustrates one embodiment of the present disclosure forgenerating a four-sided patch.

FIG. 39 illustrates an alternative embodiment of the present disclosurefor generating the four-sided patch also generated in FIG. 38.

FIG. 40 shows the notational correspondences between the geometricobjects of FIG. 38 and those of FIG. 39.

FIG. 41 shows a possible geometric configuration of FIG. 38, wherein theprofiles P₃ and P₄ of FIG. 38 are degenerate.

FIGS. 42A and 42B illustrate the movement of a marker 2002 that isconstrained to reside on the profiles curves 2003 and 2004.

FIG. 43 illustrates constraints on composed profile curves and theircorresponding isocline ribbons for providing tangent plane continuitybetween two blended surfaces S₁ and S₂.

FIG. 44 shows a profile P, associated isocline ribbons RL and RR, andvarious handles used in describing the conditions for achieving G1continuity on P.

FIG. 45 illustrates a prior art mapping of a pentagon 10 into an objectsurface 11 for the dividing of a 5-sided patch into five 3-sidedpatches.

FIG. 46 provides an illustration of the subdivision of an octagon inparameter space into eight triangles, and the mappings of each suchtriangle into object space according to the present disclosure.

FIG. 47 illustrates a surface 20 subdivided into a (green) 3-sided patchand a (red) 4-sided patch according to the present invention.

FIGS. 48A and 48B show the smoothness of the surface of FIG. 3, whereinmach bands are shown.

FIGS. 49A and 49B show isophote strips illustrating the smoothnessbetween the both patches shown in FIG. 3.

FIG. 50 uses triangular texture tiles to show the even parameterizationof the 3 and 4-sided patches of the surface 20.

FIG. 51 shows a boundary curve generated linearly from interpolatingpoints.

FIG. 52 shows a configuration for computing the parameter functions.

FIG. 53 shows a line with three points A, B, and C thereon, whereinf(u)=A+u(B−A) and g(v)=B+v(C−B), and wherein f(u) and g(v) are obviouslyC⁰ continuous at the join B, but may not be C¹ continuous at B.

FIG. 54 shows how surface continuity follows the ribbon continuity, inparticular, two ribbons QL_(i) and QR_(i) meet with G² continuity, i.e.,any curve (bold) crossing the boundary is G² continuous, then thesurfaces L and R will be G² continuous.

FIGS. 55A and 55B show, respectively, G¹ and G² geometric continuitysurfaces of a car body.

FIGS. 56A and 56B show, respectively, G¹ and G² geometric continuitysurfaces of a car hood.

FIG. 57A shows offset vectors to create curve control points where thecontrol points of the lofts are easily defined by offsetting theattribute curves control points b_(i) with symmetric displacementvectors.

FIG. 57B shows offset vectors applied across a curve to create curvecontrol points for ribbons in along the curve.

DETAILED DESCRIPTION

1. Introduction

FIG. 1 illustrates the use of an embodiment of the present disclosurefor designing a surface 62 that interpolates any two parametric surfacessuch as between the half cylinder surface 30 and the surface 34. Thatis, the surface 62 is generated via a novel surface interpolationprocess, wherein constraints on surface 62 shape are provided by thefeature curves 54, 58 and 60, and their associated novel controlgeometry (e.g., isocline ribbons). In particular, the followingconstraints are satisfied by the surface 62:

-   -   (a) one or more geometric characteristics of the surface 30        along the feature curve 54 are imposed on the surface 62,    -   (b) one or more geometric characteristics of the surface 34        along the feature curve 58 are imposed on the surface 62, and    -   (c) the surface 62 interpolates through the feature curve 60,        wherein the surface 62 tangents along the extent of curve 60 are        derived from (e.g., identical to) the isocline ribbons 61 and        63.        Thus, using a computational system as contemplated by the        present disclosure, a designer can design a surface specified in        terms of: (a) a relatively small number of carefully constructed        and positioned feature curves, and (b) the desired slope(s) of        the surface along the extent of these feature curves (via        isocline ribbons). Moreover, using a computational system as        contemplated by the present disclosure, such a designed surface        not only interpolates fairly between the feature curves but also        may obey other imposed constraints such as convexity, concavity,        and/or predetermined curvature ranges.

Additionally, a computational system according to the present disclosurecan be used to blend a surface region into an object being designed. Forexample, FIG. 2 illustrates the blending of a circular disk 66 into thecylindrical surface 30. Moreover, a computational system according tothe present disclosure can also be used to construct bosses, dimples,logos, and embossing as well as to recursively design surfaces as oneskilled in the art will come to appreciate from the disclosure herein.

At least one embodiment of a computational system according to thepresent disclosure differs from traditional approaches to computer-aideddesign (CAD) in that with a computational system according to thepresent disclosure, a desired geometric object (e.g., a surface) thatmay be created as a plurality of geometrically and computationallyunrelated patches (e.g., three-, four-, five-sided bounded surfaces),which may be subsequently pieced together in a way that satisfiescertain constraints at the boundaries between the patches. Thus, thedesired geometric object can be designed by piecing together theplurality of unrelated geometric sub-objects (subsurfaces) in a mannerthat interpolates, blends and/or trims these sub-objects so that, acrossthe boundaries and/or regions therebetween, constraints such ascontinuity, differentiability, and/or curvature are satisfied. This isfundamentally different from the traditional approaches to CAD in thatonly four-sided NURBS, Bezier, Hermite, Coons, Gordon or Booleans ofimplicit surfaces are patched together in prior art systems.

2. Blending Between Geometric Objects

A fundamental geometric object design technique of the presentdisclosure presents the blending between two parametric geometricobjects such as surfaces and, more particularly, the manner in whichsuch blending is performed. As defined in the Definitions Section above,a “parametric geometric object” (e.g. a surface) may be defined as aresult of a mapping from a (simple) coordinated geometric object(parameter space) such as a bounded plane to another (typically, morecomplex) geometric object (object space). When the parameter space is abounded plane, two coordinates or parameters (denoted u and v) may byway of example be used to uniquely identify each point in the parameterspace. When the object space is three-dimensional, for every (u,v) pointin the bounded plane parameter space, a function may associate a point(x,y,z) in the object space.

By convention, a planar parameter space is usually assumed to be theunit square, which means that both u and v vary between 0 and 1,although it is within the scope of the present disclosure to utilizeother parameter space geometries and coordinate ranges.

In one embodiment of a computational system according to the presentdisclosure, in order to blend between two parametric surfaces S₁(u,v)and S₂(u,v) each having the unit square as their parameter space, eachsurface S₁ and S₂ has associated therewith a respective blendingfunction B₁(u,v) and B₂(u,v), wherein each of the blending functionshas, for example, (0,1) as its range (as well as satisfying otherproperties given hereinbelow). Consequently, a new surface may bedefined by the following formula:S(u,v)=S ₁(u,v)·B ₁(u,v)+S ₂(u,v)·B ₂(u,v)  (1)Note that the blending functions B₁ and B₂ are typically chosen so thatthe resulting blended surface S is the same as S₁ on a boundary with S₁,and the same as S₂ on a boundary with S₂. This is achieved by devisingB₁ and B₂ so that B₁=1 and B₂=0 on the boundary with S₁ while havingB₂=1 and B₁=0 on the boundary with S₂. In FIG. 3, for example, if S₁ isthe surface 30 and S₂ the strip 34, and one boundary is the verticalline 54 of the surface 30 and the other boundary is the curve 58 on thestrip 34, then the surface 62 a is S, which runs between these twoboundaries and is tangent to S₁ and S₂ at the boundaries.2.1. The Blending Functions

Blending functions may be provided for blending between geometricobjects of various types. For example, blending functions for blendingbetween two volume filling geometric objects can be provided. However,to simplify (and clearly illustrate) the novel blending process and theassociated blending functions of the present disclosure, the discussionhere is initially limited to blending between two curves, or blendingbetween two surfaces. Accordingly, for two surfaces S₁ and S₂ to beblended together, the blending functions B₁(u,v) and B₂(u,v),respectively, are appropriately set to either 0 or 1 on the boundariesof a blended surface generated by a computational system according tothe present disclosure.

Referring to FIG. 4, wherein it is assumed that the boundaries 78 a, 78b in parameter space correspond to the profiles 54 and 58 in objectspace, for any curve 80 on the blended surface such as surface 62 a,there is a related pre-image (e.g., line 86) defined in parameter spaceas indicated. Note that for simplicity the boundaries 78 a and 78 b, andthe pre-image of curve 80 are straight, but they need not be so.

Assuming (again for simplicity) the blending functions B₁(u,v) andB₂(u,v) have their domains in the unit square (as their parameterspace), for any point (u,v) in this parameter space it is important todetermine some measure of how “close” the point (u,v) is to the boundarycurves (e.g., boundary curves 78 a and 78 b) and, more generally, to thepre-images of profile curves. Such closeness or distance-likemeasurements may be used in specifying the blending functions and/ortheir resulting values. Note that there are many ways to compute such acloseness or distance-like measurement in parameter space. For instance,if a boundary 78 (or profile pre-image) is a straight line, then such aparametric distance to a (u,v) point is easily calculated as the lengthof a perpendicular line segment to the boundary line through the point.Additional techniques for computing parametric distances are describedhereinbelow (e.g., Sections 2.3 and 2.4).

Assuming parameter space is still the 2-dimensional space of (u,v)points, a blending function B _(i) (wherein 1≤i≤N for some fixed Nnumber of boundary curves) can be computed as a function of a univariatedistance-like function B _(i)(D_(i)), where D_(i) is in turn a functionof (u,v) so that B _(i)(D_(i))=B _(i)(D_(i)(u,v))=B_(i)(u,v), whereinD_(i)(u,v) is a distance-like function to the pre-image C_(i) ⁻¹ of aboundary curve C_(i) (in object space) of a surface S. Note that suchdistance-like functions must satisfy the condition that as (u,v) getsarbitrarily close to the i^(th) boundary curve pre-image C_(i) ⁻¹ (suchas measured in conventional Euclidian distance), then D_(i)(u,v) getsarbitrarily close to zero. Examples of such blending functions B _(i)and distance-like functions D_(i) are provided hereinbelow.

Since many of the most useful blending functions B_(i) are of the form B_(i)(D_(i)), unless additional specificity is required, B_(i) will beused hereinbelow to denote both: (a) the blending function B_(i)(u,v)initially discussed above, and (b) the blending function B _(i)(D_(i))for some distance-like function D_(i). If, however, a clear distinctionis required between the blending functions of (a) and (b), the domain ofthe blending function can be used to indicate which blending function isindicated. As an aside, note that Formula (1) applies equally well forthe blending function B _(i)(D_(i)), i=1,2; that is,S(u,v)=S ₁(u,v)· B ₁(D ₁)+S ₂(u,v)· B ₂(D ₂).  (2)If a point (u,v) is close to the i^(th) pre-image boundary i=1,2, then B_(i)(D_(i)) is expected to be small and the point is mapped (into objectspace) close to the i^(th) boundary.

A good collection of blending functions B_(i) not only allows themapping, S, of a blended surface to be coincident with the desiredperimeter (profile) curves, but will do so in a manner so that theresulting blended surface between two or more such perimeter curves of,e.g. for example, different initial surfaces will preserve suchcharacteristics as the continuity of curvature with these initial twosurfaces. That is, the blended surface “heels” to each of the initialsurfaces. Also, it is preferred that the blending functions B_(i) allowthe new surface to be fair. FIG. 5 shows a graph of a pair of desirableblending functions for B_(i), i=1, 2.

For profile curves P₁, P₂ of two surfaces S₁ and S₂, wherein a blendedsurface is desired between P₁ and P₂, assume that the profiles P₁ and P₂have parametric pre-images that correspond, respectively, to u=0, u=1 ofthe unit square {(u,v)|0≤u≤1 and 0≤v≤1}, then some useful properties forblending functions B₁ and B₂ are:

-   -   (1.1) B₁=1 at u=0 and B₁=0 at u=1. B₂=0 at u=0 and B₂=1 at u=1.    -   (1.2) The derivatives B₁′ and B₂′ equal 0 wherever D_(i)(u,v)=0        and D_(i)(u,v)=1, i=1, 2. This enforces smooth (tangent        continuous) transitions between the blended surface S and the        initial surfaces S₁ and S₂. If higher order derivatives are also        zero, then higher order continuity between surfaces can be        realized, usually improving its fairness.    -   (1.3) B₁+B₂=1 for all points (u,v). This is called a “partition        of unity,” and it provides for the generation of a convex        combination of the surfaces S₁ and S₂ to which a new blended        surface abuts. Note that this tends to keep the new blended        surface from drifting too far from the initial surfaces S₁ and        S₂.

There are numerous embodiments for defining blending functions. Oneuseful embodiment is:

$\begin{matrix}{{B_{1}\left( D_{1} \right)} = {{{\cos^{2}\left( {D_{1}\frac{\prod}{4}} \right)}\mspace{14mu}{and}\mspace{14mu}{B_{2}\left( D_{2} \right)}} = {\sin^{2}\left( {D_{1}\frac{\prod}{4}} \right)}}} & (3)\end{matrix}$which gives arbitrarily high order continuity of the blending functions,which is needed to achieve the same high order continuity between theinitial blended surfaces. Another alternative is to choose polynomialfunctions with the above properties (1.1) through (1.3). For example, aquintic polynomial can be chosen with zero second derivative at D=0 andD=1, thereby providing beneficial curvature characteristics (see Section4.4).

In addition to the blending functions described herein above, thefollowing are examples of additional blending functions:

B₁(x) and B₂(x) are polynomials satisfying the following constraints:B ₁(0)=1, B′ ₁(0)=0, B(1)=0, and B′(1)=0B ₂(x)=1−B ₁(x)

Note that additional constraints regarding high order derivatives (e.g.,equal to 0 at x=0 and/or 1) may also be imposed. For example, ifB″_(i)(0)=B″_(i)(1)=0, i=1, 2, then C² continuity is attained with theobjects from which interpolating and/or blending is performed.

An example of polynomial blending functions satisfying these constraintsis:B ₁(x)=(1−x)²+5x(1−x)⁴+10x ²(1−x)³B ₂(x)=1−B ₁(x).

Note that B₁(x) may be derived as a Bezier curve with six controlpoints, P₁, . . . , P₆, as shown in FIG. 6D. Moreover, note that sinceB′ ₁(x)=−30x ²·(1−x)² andB″ ₂(x)=60x·(1−x)²−60x ²·(1−x),thatB′ ₁(1)=0, B″(0)=0, B″ ₁(1)=0, B′ ₂(0)=0, B′ ₂(1)=0, B″ ₂(1)=0, B′₁(0)=0 and B″ ₂(0)=0.

-   -   (c) Any composition of blending functions as described        hereinabove with a bijective (e.g., one-to-one and onto)        parameterization function P:[0,1]→[0,1] may be composed with a        blending function to obtain another blending function. As a        specific example, let P(x)=2c(x−x²)+x², where c is a constant        “skew” factor, then a new blending function may be construed as        B(P(x)). Thus, when c=½, P(x)=x. Moreover, as c varies, the        inflection point of the graph of P(x) moves as shown in FIGS.        6A-6C. Note that the blending function of FIG. 6B (wherein c>½)        will cause the blending curve (and/or surface or other geometric        object) to retain the geometric characteristics of the object        used for blending that corresponds to the x=0 axis in the graph        of FIG. 6B for a larger part of the surface.

To reduce the real-time design computational overhead incurred forevaluating blending functions, the values for the blending functionsmay, in one embodiment, be tabulated prior to a design session at asufficiently high resolution and stored in memory in a manner thatallows efficient indexed access to a closest approximation to the actualblending function value.

2.2. Extending Blending to N-Sided Regions

In one embodiment of a computational system according to the presentdisclosure, a novel general form for blending over a region that isbounded by each edge e_(i) of a parametric surface S_(i) (i=1, 2, . . ., N and N≥2) is the following weighted sum of points S_(i)(u_(i)(p),v_(i)(p)):

$\begin{matrix}{{S(p)} = {\sum\limits_{i = 1}^{N}{\left\lbrack {\prod\limits_{\underset{j \neq i}{j = 1}}^{N}\;{B_{j}\left( {D_{j}(p)} \right)}} \right\rbrack{S_{i}\left( {{u_{i}(p)},{v_{i}(p)}} \right)}}}} & (4)\end{matrix}$where:

-   -   (a) p is a variable denoting points in a common parameter space        for the surfaces S_(i);    -   (b) D_(j)(p) is a distance-like measurement to the pre-image of        the i^(th) edge e_(i) in the common parameter space;    -   (c) B_(j) is a blending function which is zero when D_(j) is        zero and monotonically increases as D increases; and    -   (d) u_(i) and v_(i) are parameterization functions that        transform p from the common parameter space to the (any)        intermediate parameter space for S_(i).        Note that by dividing by the sum of the products of the blending        functions, B_(j),

$\left( {{e.g.},{\sum\limits_{i = 1}^{N}\left\lbrack {\prod\limits_{\underset{j \neq i}{j = 1}}^{N}\; B_{j}} \right\rbrack}} \right)$the formula (4) can be normalized with respect to the blendingfunctions. Further note that Formula (4) resembles Formula (1) when N=2,and is in fact an extension thereof. That is, for N=2, B₁ of Formula (4)has the functional behavior of B₂ in Formula (1), and B₂ of Formula (4)has the functional behavior of B₁ in Formula (1). That is, there is asubscripting notational change between the two formulas.

As an example of Formula (4), consider the three-sided region 300 shown(in-object space) in FIG. 14. Applying Formula (4) to thereby generate asurface, S, for region 300, the following equation is obtained:S(p)=B ₂(v)B ₃(w)S ₁(u)+B ₁(u)B ₃(w)S ₂(V)+B ₁(u)₂(v)S ₃(w)  (5)where u, v and w as parameterization functions are the barycentriccoordinates of p as one skilled in the art will understand.

An alternative method to define a blended surface over N-sided (N>4)regions is provided by first applying the two-sided approach based onFormula (1) using R₁₁ and R₁₂ of FIG. 34 as S₁ and S₂, respectively inFormula (1) to thereby generate S₁ of FIG. 34. Additionally, Formula (1)is applied to the surfaces of FIG. 35, wherein S₁ and S₂ of Formula (1)are replaced by R₂₁ and R₂₂ respectively, to thereby generate S₂ of FIG.35. The two resulting surfaces S₁ and S₂ of FIGS. 34 and 35 respectivelyare, in turn, blended using Formula (2) wherein blending functions B₁and B₂ are as described hereinabove, and the corresponding D_(i) aredescribed hereinbelow. For example, given that each of the ribbons R₁₁,R₁₂, and R₂₂ have a common pre-image, the D_(i) used in Formula (2) tocompute distance-like measurements to the pre-images of the pair ofedges P₁₁, P₁₂, P₂₁, and P₂₂ (FIGS. 34 and 35) may be:

-   -   (a) For a point Pt of the (common) pre-image for S₁ of FIG. 34,        D₁(P₁)=min (D(P₁, P₁₁), D(P₁, P₁₂)) wherein D is the Euclidean        distance between P₁ and the corresponding profile P_(1i), and    -   (b) For a point P₂ of the (common) pre-image for S₂ of FIG. 35,        D₂(P₂)=min (D(P₂, P₂₁), D(P₂, P₂₂)).

Accordingly, the two surfaces St and S₂ can be blended together usingFormula (2) to obtain surface S of FIG. 36.

In another embodiment that is particularly useful for generating afour-sided blended patch, assume the following restricted but versatilescheme for defining profiles and ribbons:

-   -   (a) All handles are piecewise linear segments; and    -   (b) All blending is done with the functions B₁(x) and B₂(x) of        Formulas (3).

Moreover, referring first to FIG. 37 in describing the present patchgeneration technique, the following labeling scheme is used. For theprofile, P:

-   -   m_(L), m_(R): the left and right hand markers, respectively, of        the profile, P;    -   h_(L), h_(R): the left and right hand profile handles,        respectively, of the profile, P;    -   s_(L), s_(R): the left and right hand isocline handles,        respectively, of the profile, P;    -   b_(L), b_(R): the left and right hand ribbon tangents at the        respective left and right end points of isocline boundary R        (these ribbon tangents also being denoted as “boundary        handles”).

Using the notation of FIG. 37, surfaces S_(L) and S_(R) may be defined,wherein S_(L) is bounded by the line segments corresponding to: s_(L),h_(L), b_(L), and d_(L)=(s_(L)+b_(L))−h_(L), and S_(R) is bounded by theline segments corresponding to: s_(R), h_(R), b_(R), andd_(R)=(s_(R)+b_(R))−h_(R). In particular, S_(L) and S_(R) are known inthe art as “twisted flats,” and accordingly, S_(L) is denoted as theleft twisted flat, and S_(R) is denoted as the right twisted flat.Moreover, these surfaces may be evaluated using the following formulas(5.01a) and (5.01b):

$\begin{matrix}{{S_{L}\left( {u,v} \right)} = {\left( {{1 - v},v} \right)\begin{pmatrix}m_{L} & h_{L} \\s_{L} & b_{L}\end{pmatrix}\begin{pmatrix}{1 - u} \\u\end{pmatrix}}} & \left( {5.01a} \right)\end{matrix}$wherein the parameters u and v increase in transverse directions asillustrated by the u-direction arrow and the v-direction arrow (FIG.37).

$\begin{matrix}{{S_{R}\left( {u,v} \right)} = {\left( {{1 - v},v} \right)\begin{pmatrix}h_{R} & m_{R} \\b_{R} & s_{R}\end{pmatrix}\begin{pmatrix}{1 - u} \\u\end{pmatrix}}} & \left( {5.01b} \right)\end{matrix}$wherein the parameters u and v also increase in transverse direction,with the u-direction being the reverse direction of the u-directionarrow of FIG. 37.

Accordingly, the isocline ribbon surface S (FIG. 37) can now be definedas follows:S(u,v)=B ₂(u)S _(L)(u,v)+B ₁(u)S _(R)(u,v)  (5.02)where conveniently, the u parameter is also the distance measure neededfor B₁ and B₂ of Formulas (3). Thus, when v=0, S(u, 0) is the profile;i.e., a blend between the control handles (h_(L)−m_(L)) and(h_(R)−m_(R)). Additionally, note that when v=1, S(u, 1) is the ribbonboundary R derived as a blend of vectors (b_(L)−s_(L)) and(b_(R)−s_(R)). Also note that if b_(L) and b_(R) are translates of h_(L)and h_(R), respectively, along s_(L)−m_(L) and s_(R)−m_(R),respectively, then R is a translation of P, and such similarities maysimplify the data storage requirements of a computational systemaccording to the present disclosure.

For a plurality of isocline ribbons S₁, S₂, . . . , S_(N), wherein eachS_(i) is generated by Formula (5.02), such ribbons may now be used inthe more general N-sided surface form below, which is a variation ofFormula (4).

$\begin{matrix}{{S\left( {s,t} \right)} = \frac{\sum\limits_{i = 1}^{N}{\left( {\prod\limits_{\underset{j = 1}{j \neq i}}^{N}\;{B_{j}\left( {D_{j}\left( {s,t} \right)} \right)}} \right){s_{i}\left( {{u_{i}\left( {s,t} \right)},{v_{i}\left( {s,t} \right)}} \right)}}}{\sum\limits_{i = 1}^{N}\left( {\prod\limits_{\underset{j = 1}{j \neq i}}^{N}\;{B_{j}\left( {D_{j}\left( {s,t} \right)} \right)}} \right)}} & (5.03)\end{matrix}$Note that D_(j)(s,t), u_(i)(s,t) and v_(i)(s,t) must be defined for thisformula, i.e., the distance measure and the mappings from the generalN-side patch parameter space (in s and t) to the parameter space of theribbons S_(i) (in u and v).

For specific cases where N=2, 3, 4 and N≥5 using the blended ribbonsS_(i), notice first that Formula (5.02) for the ribbon is a special caseof Formula (5.03). For example, in Formula (5.02) the denominator is 1,the distance measure is just the u-parameter, and u and v correspondexactly to s and t. The formula for a two-sided surface is similar,except that the base surfaces are ribbons derived according to Formula(5.02) (denoted herein also a “twisted ribbons”); thus,S(u,v)=B ₂(v)S ₁(u,v)+B ₁(v)S ₂(u,v)  (5.04)which is similar to Formula (5.02), wherein the parameter u measuresdistance. It varies along the direction of the profile curve. However,in Formula (5.04), the parameter v measures distance.

Referring to FIG. 38, wherein the isocline ribbons S₁ and S₂ areparameterized as indicated by the u and v direction arrows on each ofthese ribbons, these ribbons may be used to generate a four-sided patch.The two profiles P₁ and P₂ that vary in u are blended using the twistedribbons S₁ and S₂. The other two sides P₃ and P₄ are blended profilesderived from the isocline handles; that is, P₃ is a blend (e.g., usingFormula (1)) of h_(R) ¹ and h_(L) ¹, wherein h_(R) ¹ is S₁ and h_(L) ¹is S₂ in Formula (1), and similarly, P₄ is a blend of h_(R) ² and h_(L)².

Note that the blended surface, S, of FIG. 38 has tensor product form.This can be shown by decomposing Formula (5.04) into a tensor form,wherein each of the ribbons S₁ and S₂ is derived from the Formula(5.01a) and (5.01b). That is, S₁ is a blend of S_(L) ¹ and S_(R) ¹ (FIG.38) and S₂ is a blend of S_(R) ¹ and S_(R) ².

Accordingly, the decomposition is as follows:

$\begin{matrix}\begin{matrix}{{S\left( {u,v} \right)} = {\begin{pmatrix}{B_{2}\mspace{11mu}(v)} & {B_{1}\mspace{11mu}(v)}\end{pmatrix}\begin{pmatrix}{S_{1}\left( {u,v} \right)} \\{S_{2}\left( {u,v} \right)}\end{pmatrix}}} \\{= {\begin{pmatrix}{B_{2}\mspace{11mu}(v)} & {B_{1}\mspace{11mu}(v)}\end{pmatrix}\begin{pmatrix}{S_{L}^{1}\left( {u,v} \right)} & {S_{R}^{1}\left( {u,v} \right)} \\{S_{L}^{2}\left( {u,v} \right)} & {S_{R}^{2}\left( {u,v} \right)}\end{pmatrix}\begin{pmatrix}{B_{2}(u)} \\{B_{1}(u)}\end{pmatrix}}} \\{= {\left( {{B_{2}\mspace{11mu}(v)\mspace{14mu} S_{L}^{1}}\; + {B_{1}\mspace{11mu}(v)\mspace{14mu} S_{L}^{2}\mspace{14mu} B_{2}\mspace{11mu}(v)\mspace{14mu} S_{R}^{1}}\; + {B_{1}\mspace{11mu}(v)\mspace{14mu} S_{R}^{2}}} \right)\begin{pmatrix}{B_{2}(u)} \\{B_{1}(u)}\end{pmatrix}}}\end{matrix} & (5.05)\end{matrix}$Thus, the last expression above shows that the same surface S can begenerated by first creating the twisted ribbons in the vparameterization, and then second, blending in u. However, since theroles of u and v are symmetric, the twisted ribbons may be generatedalong the u parameterization and subsequently, the blending may beperformed in v. That is, using the surfaces S_(L) ³ and S_(R) ³, S_(L) ⁴and S_(R) ⁴ of FIG. 39 gives the same surface S as in FIG. 38.Thus, in either technique for deriving S, the inputs are the same; thatis, m_(L) ^(i), m_(R) ^(i), h_(L) ^(i), h_(R) ^(i), s_(L) ^(i), s_(R)^(i), b_(L) ^(i), and b_(R) ^(i), where “i” denotes the profile P_(i)(i=1, 2, 3, 4) to which the inputs apply. Note that the correspondencesbetween the various inputs is shown in FIG. 40.

So, overall, the two-sided patch of Formula (5.04) provides a veryversatile four-sided patch. Moreover, its evaluation is also efficient.Thus, by expanding the S_(L) ^(i) and S_(R) ^(i) of Formula (5.05) usingFormulas (5.01a) and (5.01b), the following expression may be obtained:

$\begin{matrix}{{\left( {{B_{2}(v)},{B_{1}(v)}} \right)\left\lbrack {\left( {{1 - v},v} \right)\begin{pmatrix}\begin{pmatrix}1 \\L\end{pmatrix} & \begin{pmatrix}1 \\R\end{pmatrix} \\\begin{pmatrix}2 \\L\end{pmatrix} & \begin{pmatrix}2 \\R\end{pmatrix}\end{pmatrix}} \right\rbrack}\begin{pmatrix}{1 - u} \\u\end{pmatrix}\begin{pmatrix}{B_{2}(u)} \\{B_{1}(u)}\end{pmatrix}} & (5.06)\end{matrix}$where

$\begin{pmatrix}i \\L\end{pmatrix}\mspace{14mu}{and}\mspace{14mu}\begin{pmatrix}i \\R\end{pmatrix}$are the appropriate matrices from Formulas (5.01a) and (5.01b). Notethat when evaluating an instantiation of this expression, the B_(i)should probably be table driven.

The above formulation is mathematically sound, but to use it in ageometrically intuitive fashion still requires judgment on the user'spart. Thus, in certain degenerate cases, some mathematical aids are alsoin order. A common instance is where two of the profiles (e.g., P₁ andP₂) intersect each other, as in FIG. 41. This is a degenerate case sinceprofiles P₃ and P₄ (of FIG. 38) are zero length, and share end markers(i.e., m_(L) ³=m_(L) ⁴ and m_(R) ⁴=m_(L) ⁴).

Note, however, that Formula (5.04) still defines a surface S, but it iseasy to see that the surface may loop at the profile intersections. Toeliminate this looping and still maintain handle-like control at themarkers, the twisted ribbon of Formula (5.04) may be sealed by afunction of u. One function that is 1 at u=½ and 0 at u=0, is:α(u)=1−4(u−½)²  (5.07).Thus, Formula (5.01) is adjusted to be:S(u,v)=B ₂(v)α(u)S ₁(u,v)+B ₁(v)α(u)S ₂(u,v)  (5.08)Such a function (5.08) will likely remove most loops.

The ability to diminish the ribbon at the ends suggests otherapplications. A scaling function such asα₁(u)=1−u ²  (5.09)diminishes the ribbon at the u=1 end, whileα₁(u)=1−(u−1)²  (5.091)diminishes it at the u=0 end. This is an effective way to make atriangular (three-sided) surface, as one skilled in the art willunderstand.2.2.1 Bosses and Dimples from 2-Edges

The so-called “boss” feature may be obtained from a blending between twoprofile edges. The profiles may be provided as, for example, semicircles780 a and 780 b of FIG. 24 having isocline ribbons 784 a and 784 b,respectively. The ribbons 784 a and 784 b are in distinct parallelplanes. When these ribbons are blended together, a surface 786 (FIG. 24)is obtained which may be considered a boss or a dimple. Note that manyvariations, i.e., domes, rocket tips, mesas, apple tops, etc. may begenerated similarly. Moreover, if the top semicircular ribbon isrotated, the boss can be made to twist. This scheme can be used totransition between tubes, like a joint, as one skilled in the art willunderstand.

Note that in another embodiment, blending may be performed by using aneighborhood about each boundary curve (in object space) as a defaultisocline ribbon from which to blend using Formula (1) or Formula (4).Thus, by defining a value ∈>0, and taking a strip and width of eachsurface along the boundary to which the surface is to be blended, thesestrips may be used as isocline ribbons. Accordingly, the surfaceboundaries become profile curves and pre-images thereof may be used inthe Formula (1) or Formula (4).

23. Profile Curves

Since a computational system according to the present disclosure cantake a few well-positioned (object space) profile curves of varioustypes and generate a corresponding surface therethrough, as a blendedsurface according to Formula (1) above, there are two parameter spacepre-image curves for each of the surfaces S₁ and S₂ wherein these curvesare boundaries for the blending functions B₁ and B₂; that is, a curve atD_(i)=0 and at D_(i)=1 for each blending function B_(i). In fact, theremay be eight curves, as illustrated in FIG. 7, that may be used todefine a blended surface. That is, there may be two curves 78 a and 78 bin the parameter space of S₁ and two additional curves 78 c and 78 d inthe parameter space for S₂ (of course, in many cases these two parameterspaces are identified). Additionally, there are the mappings of thecurves 78 to the two surfaces 30 and 34, thereby providing thecorresponding image curves 90, 54, 58 and 91, these having respectivepre-images 78 a, 78 b, 78 c and 78 d.

Note that in the case where S₁ and S₂ have identical parameter spaces,profile 78 b is the pre-image of the profile 54. Moreover, if S₂ of 78 d(=78 b) is profile 58, then 78 b is included in the pre-image of each ofS₁, S₂ and blended surface 62.

When the present disclosure is used for surface design, a user ordesigner may think of designing a blended surface by continuouslypulling or deforming one profile curve of an initial surface to therebycreate a new surface between this initial surface and a profile curve ofanother initial surface.

Note that different types of profile or boundary curves may be used witha computational system according to the present disclosure. In someembodiments of a computational system according to the presentdisclosure, such a profile curve, C, may typically have a parametricpre-image in a parameter space, i.e. C⁻¹(s)=(u(s), v(s)) where s is aparameterization of the pre-image (e.g., 0≤s≤1). Note that parametriccurves such as C include curves having the following forms: (a) conicsincluding lines, parabolas, circles and ellipses; Bezier, Hermite andnon-uniform rational b-splines (NURBS); (b) trigonometric andexponential forms; and (c) degenerate forms like points. Additionally,note that these curve forms may be categorized orthogonally by othercharacteristics such as open, closed, degenerate and composite, as oneskilled in the art will understand.

Profile curves include curves from the following curve-type categories(2.3.1) through (2.3.5).

2.3.1. Open Curves

An “open curve” is one in which the end points of the curve are notconstrained to be coincident; e.g., the end points may be freelypositioned. Open curves are probably the most common type used by acomputational system according to the present disclosure when definingan arbitrary collection of curves (profiles) for generating a surface(in object space), wherein the surface is constrained to pass throughthe collection of curves.

2.3.2. Closed Curves

When a curve's end points match, the curve is denoted as “closed” Thismeans that the beginning point of the curve is the same as the endingpoint of the curve. Closed curves delimit regions of, e.g., a surface,and are especially useful for setting special design areas apart. Oneexample of this is the label surface for containers (described in theDefinitions Section hereinabove); e.g., surface 66 of FIG. 2. That is, alabel surface is a region that must be of a particular surface type,denoted a developable surface, so that a label applied thereto will notcrease or tear. Each such label surface is highly constrained and isusually separated from the rest of the design by a closed curve (such acurve can also serve aesthetic purposes in the design of the container).FIG. 8 shows an elliptic region 100 blended into a cylinder 108, whereinthe closed curve 110 delimits the elliptic region. A closed curve mayoften match tangencies at end points.

2.3.3. Degenerates

Several ways exist to generate a degenerate profile. In one technique,an open curve may be of zero length, or a closed curve may enclose aregion of no area. In such cases, the result is a point that may blendwith an adjacent surface. FIG. 9 shows a point blend created fromblending between a degenerate circular disk (i.e., the point labeled S₁)and the cylinder 116 (also denoted as S₂). Accordingly a simple boss 112is created on the cylinder 116. In particular, for appropriate blendingfunctions B_(i), i=1,2, a blended surface between S₁ and S₂ can beobtained using Formula (1). Moreover, since Formula (4) can be usedinstead of Formula (1), a surface can be generated that blends between aplurality of points (i.e., degenerate profiles) and an adjacent surface.FIGS. 22 and 23 show additional blends to degenerate profiles.

FIG. 22 shows a blended surface 710 that extends between the degenerateprofile (point) 714, and the circular end 718 of the cylinder 722. Inparticular, the blended surface 710 is a blending of the isoclineribbons 726 and 730, wherein the isocline ribbon 726 is a planar diskhaving the degenerate profile 714 as its center point, and the isoclineribbon 730 has the circular end 718 as its profile. Thus, letting S₁ bethe isocline ribbon 726, and S₂ be the isocline ribbon 730 in Formula(1), the distance-like measurements (in their corresponding parameterspaces) can be equated to:

-   -   (a) the radial distance from the degenerate profile 714 on the        isocline ribbon 726;    -   (b) the distance away from the profile 718 on the isocline        ribbon 730.

FIG. 23 shows another blended surface 750 that extends between thedegenerate profile (point) 754, and the planar annulus 758 having acircular curve 760 therein (and having, optionally, a central hole 762therethrough with curve 760 as its boundary). In particular, the blendedsurface 750 is a blending of the isocline ribbon 766 (for the degenerateprofile 754), and the annulus 758 (which, e.g., can optionally be anisocline ribbon to the surface 770 wherein curve 760 is a profile).Thus, letting S₁ be the isocline ribbon 766 and S₂ be the annulus 758,the distance-like measurements (in their corresponding parameter spaces)can be equated to:

-   -   (a) the radial distance from the degenerate profile 754 on the        isocline ribbon 766;    -   (b) the distance away from the curve 760 on the annulus 758.

2.3.4. Composite Curves

The novel geometric design techniques of the present disclosure can alsobe utilized with composite curves. Composite curves are general curveforms that include other curves as sub-curves, wherein the sub-curvesmay cross or may kink, e.g., at endpoints. In utilizing composite curvesas, e.g., profiles, the definition of a distance-like measurement for acomposite curve is important. FIG. 10 shows a composite curve 120 thatincludes two crossing sub-curves 124 and 128. However, such compositecurves can also have their sub-curves strung end-to-end.

Assuming the sub-curves C_(j), j=1, 2 . . . , N of a composite curveCare parameterized and have a common parameter space, a distance formula(in parameter space) for determining a distance-like measurement D tothe pre-images of the sub-curves C_(j) is:D(p)=D _(N)(P′), andD _(k)(P)=d _(k)(P)+D _(k-1)(P)−[d _(k) ²(P)+D _(k-1) ²(P)]^(1/2)  (5.5)where k=2, . . . , N and D₁(p)=d₁(P)=a distance measurement between Pand C₁, and D_(k)(P)=a distance measurement between P and C_(K). Thus,D(p) can be used as the input to a blending function, B(D), for blendingone or more surfaces to the composite curve, C.2.3.5. Trimming Curve

A computational system according to the present disclosure allows asurface to be “trimmed,” wherein trimming refers to a process forconstraining or delimiting a surface to one side of a particularboundary curve (also denoted a trim curve). In particular, forparameterized surfaces, the pre-image of a trim curve, e.g., in the(u,v) parameter space of the surface, identifies the extent of thepre-image of the surface to remain after a trimming operation. A trimcurve may be a profile curve, and the desired trimmed surface is thatpart of the original untrimmed surface that typically lies on only oneside of the trim curve. An example is shown in FIG. 11, wherein theoriginal untrimmed surface is the generally rectangular portion 130. Therounded surface 134 is a “label” surface that is trimmed to the curve138 from the original surface 130. Note the trim profile 138 may have anassociated isocline ribbon (not shown) for one or more adjacent surfaces(e.g., surface 142) that heel to an isocline ribbon at the trimmingprofile 138. The use of isoclines for modifying the shape of suchadjacent surfaces is an important technique in creating a smoothtransition from the adjacent surfaces to a trimmed surface.

Note that a computational system according to the present disclosure mayinclude a trimming technique to create a hole in a geometric object. Byextruding a depression in a front surface of the geometric objectthrough a back surface of the object, and then trimming the frontsurface to exclude the corresponding portion on the back surface, a holecan be constructed that can be used, e.g., as a handle of a container.

2.4. Distance Metrics

Some techniques for computing distance-like measurements have alreadybeen provided hereinabove. In this section, additional such techniquesare described. The efficiency in computing how close a point inparameter space is to one or more particular geometric object pre-images(curves) in parameter space can substantially impact the performance ofa geometric design and modeling embodiment of a computational systemaccording to the present disclosure. In general, for computing suchdistance-like measurements (these being, in general, a monotonicfunction of the conventional Euclidean distance metric) in parameterspace between points and curves, there is a trade-off between thecomplexity of the curve and how efficiently such measurements can beevaluated. In general, the simpler the curve, the faster such distancescan be determined. As an aside, it should be noted that for a parameterspace curve and its image curve (in object space), these curves need notbe of the same computational type (e.g., polynomial, transcendental,open, closed, etc.). Indeed, a parameter space curve may be quite simpleand still be the pre-image of a complicated surface curve in objectspace. For example, the parameter space curve corresponding to theBezier curve 58 in FIG. 1 may be a straight line. By keeping theparameter space curve as simple as possible, fast distance computationsare possible.

2.4.1. Parametric Distance Calculations for Blending

This section describes a variety of methods for calculating adistance-like measurement (more generally, a monotonic function of theconventional Euclidean distance metric) to a number of candidateparameter space curves, wherein the methods are listed in a roughlyincreasing order of computational complexity.

Assume a blended surface is to be generated between two profile curvesP₁ and P₂, each having isocline ribbons, wherein each ribbon isparametric and has, e.g., the planar unit square [0,1]×[0,1] as thecommon parameter space for the ribbons. One distance-like functioncapable of being used for blending is a function that is dependent ononly one or the other coordinate of points represented by the coordinatepairs (u,v) in the common parameter space. That is, assuming the profilecurves P₁ and P₂ of the isocline ribbons are such that their pre-imagesare the vertical lines u=k₁ and u=k₂ for 0≤k₁≤k₂≤1, then thecorresponding distance-like functions can be D₁(u,v)=(u−k)/(k₁−k₂) andD₂(u,v)=(u−k₁)/(k₂−k₁). Moreover, if the pre-images are the parameterspace bounding vertical lines u=0 and u=1 (i.e., k₁=0 and k₂=1), thenthe corresponding distance-like function can be D₁(u,v)=1−u andD₂(u,v)=u, and accordingly such simple distance-like functions can becomputed very efficiently.

In order to maintain the desired simplicity in parametric distancecomputations when there are pre-images to more than two profiles forblending therebetween, three methods can be employed for computingparametric distance-like measurements. Each of the three methods is nowdescribed.

A triangular domain in parameter space bounded by, e.g., three profilecurve pre-images (that are also curves) can be parameterized withrespect to the vertices v₁, v₂ and v₃ of the triangular domain usingthree (real valued) parameters r, s and t with the additional constraintthat r+s+t=l. In other words, a point p in the triangular domain havingthe vertices v₁, v₂ and v₃ can be represented as p=r*v₁+s*v₂+t*v₃. Ther, s, t parameters are called “barycentric coordinates” and are used forthree-sided surfaces such as the surface 300 of FIG. 14 in parameterspace.

Domains in parameter space that are bounded by the pre-images of fourprofiles (denoted the four-sided case) can be a simple extension of thedomain having bounds on two opposing sides (denoted the two-sided case).In the two-sided case, if parameterized properly, only one parameter, u,need be used in the distance-like function computation. In thefour-sided case, both parameters u and v may be employed, as well astheir complements (assuming an appropriate representation such as theunit square in parameter space). Thus the distance to the four profilepre-image boundaries in parameter space can be u, v, 1−u, and 1−v (i.e.,assuming the pre-images of the profiles are u=0, v=0, v=1, u=1).

To determine barycentric coordinates for parametric space domains,assuming the pre-images of the profiles are line segments that form apolygon, the approach illustrated in FIG. 12 (illustrated for afive-sided polygon 148 having vertices v₁, v₂, v₃, v₄ and v₅) may beutilized, wherein the profile pre-images are the heavy lines labeled 149a through 149 e. To determine a distance-like function, first, stellate,i.e., make a star from, the pre-image polygon 148 by extending each ofthe sides 149 a through 149 e of the polygon until they intersect withanother extended side having a side 149 therebetween. Thus, theintersection points 150 a through 150 e are determined in the five-sidedcase of FIG. 12. Subsequently, the line segments 152 a through 152 efrom the corresponding points 150 a through 150 e to a point p in thepolygon may be constructed. The resulting distance-like measurements arethe lengths of the line segments 153 a through 153 e from p to the sides149 a through 149 e of the polygon 148. Accordingly, the distance from pto the i^(th) side 149 (i=a, b c, d, e) of the polygon 148 is thedistance along the i^(th) line segment 153 from p to the boundary edgeof the polygon 148. Note that by dividing each resulting distance-likemeasurement by the sum of all the distance-like measurements to thepoint p, the distance-like measurements can be normalized.

2.4.2. Straight Line

A straight line is represented by the equation, au+bv=c, wherein a, b, care constants. A convenient (unsigned) distance to a line is obtained byD(u,v)=|(a,b)((u,v)−c|.  (6)For a more intuitive version that corresponds to Euclidean distance,Formula (6) can be normalized to obtainD(u,v)=|(a,b)((u,v)−c|/(a ² +b ²),  (7)by dividing by the length of the gradient.2.4.3. Conics

Conics include parabolas, hyperbolas and ellipses. The general form of aconic isAu ² +Buv+Cu ² +Du+Ev+F=0.Its unsigned distance can be computed by

$\begin{matrix}{{D\left( {u,v} \right)} = {{{\left( {u,v} \right) \cdot \begin{bmatrix}A & \frac{B}{2} \\\frac{B}{2} & C\end{bmatrix} \cdot \begin{pmatrix}u \\v\end{pmatrix}} + {\begin{pmatrix}D \\E\end{pmatrix} \cdot \left( {u,v} \right)} + F}}} & (8)\end{matrix}$This can also be normalized by dividing through by the length of thegradient of the function to make a more suitable distance-like function,which is Euclidean in the case of the circle. Note that Farin inIntroduction to Curves and Surfaces, Academic Press, 4th ed., 1996,(incorporated herein by reference) gives the conversion between theimplicit form above and a rational parametric form. Thus, Formula (8)can be used regardless of whether the conic is represented implicitly orparametrically.2.4.4. Polynomial Curves, both Parametric and Implicit

Assume that a parametric curve has been converted to a Bezier form as,for example, is described in the Farin reference cited hereinabove.Vaishnav in Blending Parametric Objects by Implicit Techniques, Proc.ACM Solid Modeling Conf., May 1993 (incorporated herein by reference)gives a method to change a curve from a parametric curve to an implicitcurve numerically, wherein distance is implicitly measured in objectspace by offsetting the curve in a given direction that is based on someheuristics about how the offset is to be computed. The value of theoffset distance that forces the offset to go through the point is thedistance measurement for that point. In particular, for a Bezier curve,this distance-like measurement may be worthwhile in that it is robust(i.e., not ill-conditioned) and reasonably fast to evaluate, requiringonly two or three Newton-Raphson iterations on average, as one skilledin the art will understand. While this may be an order of magnitudeslower than computing a distance measurement of a conic representation,it is much faster than the traditional method of computing aperpendicular distance, which is also unstable.

2.4.5. Piecewise Parametric Curves

The present disclosure also includes a novel technique for computing adistance-like measurement on complex curves in parameter space.

Referring to FIG. 13, assume that both boundary curves 156 a and 156 bare in the unit square parametric space 158 and are piecewise parametricpolynomial curves that have corresponding sub-curves 160 a, 160 b of thesame degree n. By connecting end points of the corresponding sub-curveswith line segments 164 (i.e., degree one curves), degree n by 1 Bezierpatches 168 can be constructed in the unit square representation ofparameter space 158. Note that each patch 168 can be considered as asecond parameter space unto itself having coordinates (s,t) wherein:

-   -   (a) for two Bezier sub-curves 160 a and 160 b (denoted herein        b₁(t) and b₂(t), 0≤t≤1), each value, t₀, oft corresponds to a        line segment, L_(t) _(o) , between b₁(t₀) and b₂(t₀), and    -   (b) the L line segment is parameterized by s so that L_(t) _(o)        (s)∈b(t₀) when s=0 and L_(t) ₀ (s)∈b₂(t₀) when s=1, wherein s        varies proportionally with the distance between b₁(t₀) and        b₂(t₀) when 0<s<1. Accordingly, if the distance-like measurement        between the curves b₁(t) and b₂(t) (and/or patch bounding line        segments 164) is computed in the second parameter space, then        for any (u,v) point interior to the patch, it is necessary to        find the corresponding (s,t) point relative to the boundary        curves of such a patch that can then be evaluated for        determining the distance-like measurement. Since s is the linear        parameter (corresponding to the distance of a point between the        two corresponding sub-curves 160 a and 160 b that are joined at        their endpoints by the same two segments 164), simple functions        f₁(s) and f₂(s), such as f₁(s)=s and f₂(s)=1−s, can serve as        distance functions to b₁(t) and b₂(t), respectively. Note that        the parameters u and v can both be represented as Bezier        functions of s and t. In particular, to convert from (s,t)        coordinates to (u,v) parameter space coordinates, a Newton type        algorithm may be used, as one skilled in the art will        understand.

Another approach for determining the distance-like measurement, in somecircumstances, is to evaluate such patches 168 with a “forwardalgorithm.” That is, referring to FIG. 25, an object space blendedsurface 808 that blends between, e.g., profiles 812 and 816 (havingisocline ribbons 820 and 824, respectively, to which the surface 808heels) is shown. The profile 812 has as its pre-image curve 160 a (inparameter space 158), and the profile 816 has as its pre-image curve 160b (in parameter space 158), wherein the portion of parameter space 158for surface 808 is the patch 168. An additional parameter space 828 in sand t can be considered as a pre-image parameter space for the parameterspace 158 wherein the pre-image of curve 160 a is the vertical linesegment at s=0, and the pre-image of curve 160 b is the vertical linesegment at s=1. If a sufficiently dense set of points 830 denoted by“x”s in the additional parameter space 828 is used to evaluate points(u,v) in patch 168 (e.g., by determining a closest point 830), then thecorresponding points p(u,v) on a blended surface 808 can be efficientlycomputed since the distance-like functions to pre-image curves 160 a and160 b can be D₁(u(s,t), v(s,t))=s and D₂(u(s,t), v(s,t))=1−s,respectively. This approach will generate the blended surface easily andquickly. Note, if the surface 808 does not require a subsequent trimmingoperation, this method is particularly attractive.

3. Blending Programs

FIG. 17 shows a block diagram of the typical flow of design constructionoperations performed by a user of a computational system according tothe present disclosure. Thus, profile handles may be needed to constructan associated profile, and the profile is required to construct theassociated isocline ribbon, and the isocline ribbon may be required toobtain the desired shape of the associated object (e.g., a surface),which, in turn, is required to construct the desired geometric model.

FIGS. 26 through 30 provide a high level description of the processingperformed by an embodiment of the present disclosure that enables thenovel real-time manipulation of the shape of geometric objectrepresentations so that a user can more efficiently and directly expresshis/her design intent. Moreover, it should be noted that a fundamentaltenet of the present disclosure presents a paradigm shift away fromtypical CAD systems. That is, in a typical CAD system the user mustsupply input for changing or modifying a geometric object andsubsequently request processing of the input to commence. Thus, the userreceives feedback about his/her design at discrete user requested times.Instead, with a computational system according to the presentdisclosure, updates may be processed in real-time immediately upon inputwithout the user explicitly indicating that update processing is to beperformed. Accordingly, a user of a computational system according tothe present disclosure can efficiently perform iterative approximationsto a geometric object being designed without requiring the user toprecisely calculate geometric characteristics for substantially allportions of the object. In particular, this can have substantialefficiency benefits in that for many geometrically designed objects(including machined parts), substantial portions of such objects may besatisfactorily designed using a wide range of geometrically shapedobjects. Accordingly, a computational system according to the presentdisclosure allows many of these geometric objects to be designed withoutthe user having to needlessly specify precision in those portions of theobject where the precision is unnecessary.

In FIG. 26, the steps are shown for computing an interpolating curveaccording to a computational system according to the present disclosureusing a one-dimensional variation of Formula (1) discussed in Section 2hereinabove. Accordingly, in step 1004, the end points and tangents atthe end points for the interpolating curve, C(u), to be generated areobtained. In particular, the end points of this curve are assigned tothe variables PT1 and PT2. Additionally, direction vectors for theinterpolating curve C(u) at the points PT1 and PT2 are assigned to thevariables TAN1 and TAN2, respectively. Note that PT1, PT2, TAN1 and TAN2can be supplied in a variety of ways. For example, one or more of thesevariables can have values assigned by a user and/or one or more may bederived from other geometric object representations available to theuser (e.g., another curve, surface or solid representation). Inparticular, the direction tangent vectors denoted by TAN1 and TAN2 maybe determined automatically according to a parameterization of ageometric object (e.g., a surface) upon which the points PT1 and PT2reside.

In steps 1008 and 1012, the blending functions B₁ and B₂ are selected asdiscussed in Section (2.1) hereinabove. Note, however, that the blendingfunctions provided may be defaulted to a particular pair of blendingfunctions so that a user may not need to explicitly specify them.However, it is also within the scope of the present disclosure that suchblending functions may be specifically selected by the user. In thisregard, note that since the present disclosure presents a computationalsystem that is intended to express the user's geometric design intent,there may be a wide variety of blending functions that are acceptablesince typically a user's intent is often adequately expressed withoutthe user purposefully determining highly precise input. That is, it isbelieved that a wide variety of blending functions may be acceptable foriterative approximation of a final geometric design since progressivelyfiner detail can be provided by generating and/or modifyingprogressively smaller portions of the geometric object being designedusing substantially the same blending functions. Said another way, sincethe present disclosure supports both the entering of precise (geometricor otherwise) constraints as well as the iterative expression of theuser's intent at progressively higher magnifications, the high precisionand/or small scale design features may be incorporated into a userdesign only where necessary.

In step 1016, the interpolating curve, C(u), is computed using avariation of Formula (2) applied to a one-dimensional parameter space.An example of an interpolating curve, C(u), with points PT1, PT2, andvectors TAN1 and TAN2 identified, is illustrated in FIG. 32.

In FIG. 27, a flowchart is provided showing the steps performed whenconstructing an approximation to an isocline boundary R(u) for an object(e.g., a surface) S, wherein the points PT1 and PT2 delimit a profilecurve corresponding to the isocline ribbon boundary approximation to begenerated. In particular, the approximate isocline ribbon boundarygenerated by this flowchart is intended to approximately satisfy theisocline ribbon boundary definition in the Definitions Sectionhereinabove. More precisely, the isocline ribbon boundary approximationdetermined by the present flowchart will tend to match the isoclineribbon boundary definition for a portion of the object S between PT1 andPT2 depending on, e.g., how smooth the object is along the profile curvegenerated between PT1 and PT2. That is, the smoother (reduced curvaturefluctuations), the more likely the match. Accordingly, in step 1104 ofFIG. 27, the curve interpolation program represented in FIG. 26 isinvoked with PT1, PT2 and their respective tangents TAN1 and TAN2 forthe object (surface) S. Thus, an interpolating curve, C(u), is returnedthat is an approximation to the contour of S adjacent to this curve.

In steps 1108 and 1112, a (traverse) tangent (i.e., a picket) along theparameterization of the object S at each of the points PT1 and PT2 isdetermined, and assigned to the variables PICKET1 and PICKET2,respectively. Note that typically the pickets, PICKET1 and PICKET2, willbe transverse to the vectors TAN1 and TAN2, although this need not beso. Subsequently, in steps 1116 and 1120, the isocline ribbon pointscorresponding to PT1 and PT2 are determined and assigned to thevariables, RIBBON_PT1 and RIBBON_PT2, respectively. Then, in step 1124,the curve interpolation program of FIG. 26 is again invoked with thevalues RIBBON_PT1, RIBBON_PT2, TAN1 and TAN2 to thereby generate theisocline ribbon boundary approximation, R(u). It is worthwhile to notethat in some cases, the isocline ribbon approximation bounded by theinterpolating (profile) curve C(u), the corresponding pickets (PICKET1and PICKET2), and the newly generated isocline boundary R(u) does notnecessarily form a surface. In fact, the curves, C(u) and R(u) may besubstantially coincident (e.g., if PICKET1 is identical to TAN1, andPICKET2 is identical to TAN2).

In FIGS. 28A and 28B, a flowchart for a program is provided forconstructing a more precise isocline ribbon boundary than theapproximation resulting from FIG. 27. In particular, in the flowchart ofFIGS. 28A and 28B, the program of FIG. 27 is repeatedly and adaptivelyinvoked according to the variation in the object (e.g., surface) S alongthe path of the profile curve provided thereon. Accordingly, in step1204 of FIG. 28A, a sequence of one or more markers M_(i), i=1, 2, . . .N, N≥1, is assigned to the variable MARKER_SET, wherein these markersare on the surface, S, and the markers are ordered according to theirdesired occurrences along a profile curve to be generated. Note that inone typical embodiment, the markers are generally provided (e.g.constructed and/or selected) by a user. Moreover, for the presentdiscussion, it is assumed that the tangents to the surface Scorresponding to the markers M_(i) are tangents to S entered by theuser. However, it is within the scope of the present disclosure thatsuch tangent vectors may be provided automatically by, for example,determining a tangent of the direction of a parameterization of thesurface S.

In step 1208 of FIG. 28A, the first marker in the set, MARKER_SET, isassigned to the variable, MARKER1. Subsequently, in step 1212, adetermination is made as to whether there is an additional marker inMARKER_SET. If so, then in step 1216, the variable, INTRVL, is assigneda parametric increment value for incrementally selecting points on theprofile curve(s) to be subsequently generated hereinbelow. In oneembodiment, INTRVL may be assigned a value in the range greater than orequal to approximately 10⁻³ to 10⁻⁶.

In step 1220, the variable, MARKER2, is assigned the value of the nextmarker in MARKER_SET. Subsequently, in step 1224, the curveinterpolation program of FIG. 26 is invoked with MARKER1 and MARKER2(and their corresponding user-identified tangent vectors) for therebyobtaining an interpolating curve, C_(j)(u) between the two markers(where j=1, 2, . . . , depending on the number of times this step isperformed). Then in step 1228, an isocline boundary approximation isdetermined according to FIG. 27 using the values of MARKER1, MARKER2 andthe interpolating curve, C_(j)(u), for obtaining the isocline boundaryapproximation curve, R_(j)(u).

Subsequently, in step 1240, the variable, u_VAL, is assigned the initialdefault value INTRVL for selecting points on the curves, C_(j)(u) andR_(j)(u). Following this, in step 1244, the variable INCRMT_PT isassigned the point corresponding to C_(j)(u_VAL). Subsequently, in step1245, the variable, S_PT, is assigned a point on S that is “closest” tothe point C_(j)(u_VAL) More precisely, assuming S does not fold backupon itself closer than ε>0, for some ε, a point on S is selected thatis in a neighborhood less than ε of C_(j). Note that since C_(j)(u_VAL)may not be on S, by setting the value of INTRVL so that this variable'svalue corresponds to a maximum length along the interpolating curveC_(j) of no more than one-half of any surface S undulation traversed bythis curve, then it is believed that the interpolating curve C_(j) willeffectively follow or be coincident with the surface S. Subsequently, instep 1246, a determination is made as to whether the point INCRMT_PT iswithin a predetermined distance of S_PT (e.g., the predetermineddistance may be in the range of 10⁻³ to 10⁻⁶). In particular, thepredetermined distance may be user set and/or defaulted to a systemvalue that is changeable depending upon the application to which thepresent disclosure is applied. Accordingly, assuming that INCRMT_PT andS_PT are within the predetermined distance, then step 1248 isencountered wherein the point R_(j)(u_VAL) on the isocline boundaryapproximation is determined and assigned to the variable, RIBBON_PT.Subsequently, in step 1252, an approximation to an isocline picket atC_(j)(u_VAL) is determined and assigned to the variable, PICKET.

In step 1254, the tangent to the surface (more generally, object) S atthe point C_(j)(u_VAL) is determined and assigned to the variable,INCRMT_TAN, this tangent being in the direction of the parameterizationof S.

In step 1256, a determination is made as to whether the vectors,INCRMT_TAN and PICKET are sufficiently close to one another (e.g.,within one screen pixel). If so, then a subsequent new point on theinterpolating curve C_(j) is determined by incrementing the value ofu_VAL in step 1264. Following this, in step 1268, a determination ismade as to whether the end of the interpolating curve, C_(j)(u), hasbeen reached or passed. Note that the assumption here is that 0≤u≤1.Accordingly, if u_VAL is less than 1, then step 1244 is againencountered, and some or all of the steps through 1256 are performed indetermining whether the isocline ribbon point approximation,R_(j)(u_VAL), is close enough to the actual ribbon point astheoretically defined in the Definitions Section hereinabove.

Referring again to step 1246, note that if INCRMT_PT is not close enoughto S, then an interpolating curve more finely identified with the actualpoints of S is determined. That is, the point, S_PT, is made into amarker and inserted into MARKER_SET, thereby causing new interpolatingribbon curves, C_(j)(u) and R_(j)(u) to be generated (in steps 1224 and1228) that will deviate less from S (assuming S is continuouslydifferentiable). That is, step 1272 is performed wherein a marker isgenerated for the point, S_PT, and this new marker is inserted intoMARKER_SET between the current marker values for MARKER1 and MARKER2.Subsequently, the marker currently denoted by MARKER2 is flagged asunused (step 1276), and in step 1280, the most recently constructedinterpolating curve C_(j)(u) and any associated ribbon boundary curveR_(j)(u) are deleted. Then, step 1220 and subsequent steps are againperformed for determining new interpolating and ribbon boundary curves,C_(j)(u) and R_(j)(u).

Note that steps 1272 through 1280 and step 1220 are also performed if instep 1256, INCRMT_TAN and PICKET are not determined to be sufficientlyclose to one another in the object space of S.

Referring again to step 1268, if the end of the interpolating curve,C_(j)(u), has been reached or passed, then it is assumed that C_(j)(u)is a sufficiently close approximation to points on S (between MARKER1and MARKER2), and R_(j)(u) is sufficiently close to an isocline ribbonfor these points on S. Thus, if there are additional markers wherein aninterpolating curve C_(j)(u) and corresponding ribbon approximationR_(j)(u) has not been determined, then the next pair of consecutivemarkers (of the marker ordering) in MARKER_SET is determined and variousof the steps 1220 and beyond are performed. That is, in step 1284,MARKER1 is assigned the value of MARKER2, and in 1288, a determinationis made as to whether there is a next unused marker in MARKER_SET. Ifso, then variations of the steps 1220 and beyond are performed asdescribed above. Alternatively, if all markers have been designated asused, then in step 1292 the resulting curves C_(j)(u), R_(j)(u), foreach used j=1, 2, . . . , are graphically displayed and stored forsubsequent retrieval. Note that the profile curves C_(j)(u) may beoptionally reparameterized so that these curves may be parameterizedcollectively as a single curve, {tilde over (C)}(u), with{tilde over (C)}(0)=C ₁(0) and {tilde over (C)}(1)=C _(N)(1).

FIGS. 29 and 30 provide high-level descriptions of flowcharts formodifying one or more surfaces (more generally geometric objects) bymodifying isocline handles, ribbon tangents, and their associatedisocline ribbons. In particular, for simplicity, the flowcharts of thesefigures assume that there is a composite surface, S₀, that is provided(at least in part) by one or more subsurfaces, S_(i), i=1,2, . . . N,N≥1, where these subsurfaces S_(i) are connected to one another (e.g.,patched together) along common boundaries so that S₀ does not havedisconnected portions. Accordingly, given such a composite surface, S₀,the flowcharts of FIGS. 29 and 30 can be described at a high level asfollows. In FIG. 29, an isocline handle and/or a ribbon tangent havingat least one geometric characteristic (e.g. length, direction,curvature, etc.) to be changed is determined along with the subsurfacesS_(i) that are to be modified to reflect the isocline handle and/orribbon tangent changes. Subsequently, in the flowchart of FIG. 30, themodifications to the subsurfaces are computed and displayed in real-timeas the user enters the modifications to the selected isocline handleand/or ribbon tangent. Note that the computing of surface (moregenerally geometric object) modifications in real-time has notheretofore been feasible for surfaces in higher dimensional geometricobjects in that the computational overhead has been too great.Accordingly, a computational system according to the present disclosurehas reduced this overhead by providing a novel technique of computingblended surfaces which is very efficient and which generates surfacesthat are fair.

The following is a more detailed description of FIGS. 29A and 29B. Instep 1400, if there are not profiles and isocline ribbons correspondingto the entire boundary of each subsurface S_(i), then make profiles andisocline ribbons that approximate the entire boundary of each subsurfaceS_(i). Note that this may be performed using the program of theflowchart of FIG. 28. In step 1404, the isocline handles and ribbontangents corresponding to markers on the surface S₀ are graphicallydisplayed to the user. In step 1408, a determination is made as towhether the user has requested to add one or more additional isoclineribbons to the surface S₀, or extend an existing isocline ribbon havingits profile curve on S₀. If the user has made such a request, then step1412 is performed to assure that in addition to any other markers addedby the user, markers are added: (a) whenever a profile contacts aboundary of a subsurface S_(i), and (b) so that profile curves will beextended in a manner that terminates each one on a boundary of asubsurface S_(i). Moreover, additional markers may be also added atintersections of curve profiles. Thus, for these latter markers, theremay be two distinct ribbon tangents associated therewith (i.e., one foreach subsurface).

Subsequently, in step 1416, the program of FIG. 28 is invoked with eachS_(i), i=1, 2, . . . N for thereby obtaining the desired additionalprofiles and isocline boundaries. As an aside, note that FIG. 28 needonly be invoked with the subsurfaces S_(i) to which new markers areadded.

In step 1420 following step 1416, all newly added isocline handles andribbon tangents are displayed. Note that in some embodiments, only theisocline handles are displayed initially, and the user is able toselectively display the ribbon tangents as desired.

Subsequently, in step 1424, a determination is made as to whether theuser has requested to add one or more additional markers within existingprofiles. If so, then the additional new markers are added and at leastthe corresponding new isocline handles are determined for these newmarkers. As an aside, note that in one embodiment of the presentdisclosure, when a new marker is added to an existing profile, theprofile will change somewhat since it is now exactly identical to thesurface S₀ at another point and the interpolating curve generated (viaFIG. 26) between consecutive markers of a profile is now generated usingthe newly added marker. Accordingly, a profile with one or moreadditional markers should, in general, conform more closely to the shapeof the adjacent portions of the surface S₀.

Subsequently, in step 1432, the additional new markers and optionally,their corresponding isocline handles and ribbon tangents, aregraphically displayed to the user.

Note that it is not necessary for steps 1408 through 1420, and steps1424 through 1432 to be performed sequentially. One skilled in the artof computer user interface design will understand that with event drivenuser interfaces, the processing of each new marker can be performedindividually and displayed prior to obtaining a next new marker locationfrom the user. Thus, consecutive executions of the steps 1408 through1420 may be interleaved with one or more executions of the steps 1424through 1432.

In step 1436, a determination is made as to whether an isocline handleand/or a ribbon tangent is selected by the user for modification. Notethat the identifier, ISO, will be used to denote the isocline handleand/or the ribbon tangent to be modified.

In step 1440, the marker corresponding to ISO is determined and accessthereto is provided via the variable, MRKR. Subsequently, in step 1444,the collection of one or more subsurfaces S₁, . . . , S_(N) adjacent toMRKR are determined and access to these adjacent subsurfaces is providedby the variable, ADJ_SURFACES.

In steps 1448 through 1460, boundary representations of portions of thesubsurfaces, S_(i), adjacent MRKR are determined (step 1452) andinserted into a collection of surface boundary representations denotedMOD_SET (step 1456). In particular, for each of the subsurfaces inADJ_SURFACES, a data representation of the boundary of the smallestportion of the subsurface that is adjacent to MRKR, and that is boundedby isocline ribbons is entered into the set, MOD_SET.

Finally, in step 1464, the program of FIG. 30 is invoked for modifying,in real-time as the user modifies ISO, the portion of S₀ within theboundary representations contained in MOD_SET. In particular, theprogram of FIG. 30 is invoked with the values, MRKR and MOD_SET.

In the flowchart of FIG. 30, the high-level steps are shown formodifying in real-time the surface portions identified by the surfaceboundary representations in MOD_SET, wherein these surface portions areadjacent to the marker, MRKR. Accordingly, in step 1504, the first(next) modified version of the isocline handle and/or ribbon tangentcorresponding to the marker, MRKR, is obtained and assigned to, ISO.Subsequently, in step 1508, all isocline ribbons containing the modifiedisocline handle and/or ribbon tangent of ISO are regenerated to reflectthe most recent modification requested by the user. Note that this isperformed using the one-dimensional version of Formula (1), andmodifying each such isocline ribbon along its extent between MRKR andthe adjacent markers on each isocline ribbon containing MRKR.

Subsequently, in step 1512, the first (next) boundary representation inMOD_SET is assigned to the variable, B. Then in step 1516, the set ofisocline ribbons for the (profile) boundary segments contained in B areassigned to the variable, R. Note that R includes at least one isoclineribbon containing the marker, MRKR.

In step 1520, a blended surface is generated that is delimited by theprofiles of the isocline ribbons of R. The formula used in this step issimilar to Formula (4). However, there are additional functions,Q_(i)(p), provided in the present formula. Note that, in general, theportion of a parameter space used in generating the surface, S (of whichS(p) is a point), of this step may have two, three, four, five or moresides (profile pre-images) that also have isocline ribbon pre-images.Thus, a translation function, Q_(i)(p), is provided for each isoclineribbon R_(i) of R (wherein for the points p in the parameter space thatare in the interior, I, to the pre-images of the profiles, P_(i), forthe isoclines R_(i) of R) it is desirable that these points p betranslated into points in the parameter space for R_(i) so that acorresponding point in the object space of the isocline ribbon R_(i) canbe determined and used in the blending function of the present step.Note that the translation functions, Q_(i)(p), preferably satisfy atleast the following constraints:

-   -   (a) Q_(i)(p) is a continuous function for continuous surfaces;

$\begin{matrix}{{(b)\begin{matrix}{LIMIT} \\{p->{Q_{i}^{- 1}\left( {u,0} \right)}}\end{matrix}\left( {Q_{i}(p)} \right)} = \left( {u,0} \right)} & \;\end{matrix}$That is, when a sequence of points in I converges to the pre-image ofthe profile point P_(i)(u) (i.e., Q_(j) ⁻¹(u, 0)), then Q_(i)(p)converges to the isocline ribbon parameter space point (u, 0).

Subsequently, in step 1524, the surface S is displayed, and in step 1528a determination is made as to whether there is an additional boundaryrepresentation in MOD_SET for generating an additional blended surfaceS. If so, then step 1512 is again performed. Alternatively, if there areno further boundary representations, then in step 1532, a determinationis made as to whether there is an additional user modification of theisocline handle and/or ribbon tangent corresponding with MRKR. If thereis, then at least the steps 1504 through 1528 are again performed. Notethat the steps of FIG. 30 can be sufficiently efficiently performed sothat incremental real-time changes in the isocline handle and/or ribbontangent for MRKR designated by the user can be displayed as the usercontinuously modifies this isocline handle and/or ribbon tangent.

4. A Geometric Design User Interface

The general principles described above form a basis for a novel userinterface for computer aided geometric design.

In one user interface embodiment for the present disclosure, a userinterface may be provided for defining isoclines. Using such aninterface, a designer may, for example, require that an isocline beperpendicular to a given light direction along an entire profile curveso as to create a reflection line, as one skilled in the art willunderstand. More generally, the novel user interface may allow forvarious constraints to be input for generating isocline ribbons,isocline handles and/or ribbon tangents that satisfy such constraints.In particular, the user interface allows for global constraints such aslight direction, curvature, tangency, level contours, dihedral anglefunctions with a plane, etc., as one skilled in the art will appreciate.

In one embodiment of the user interface, the user will start with agiven geometric object, for example a cylinder. The user may theninscribe a profile curve on the cylinder by placing markers at variouspoints on the cylinder. The profile tangents and/or isocline handles maybe defaulted by adopting the slope information from the cylinder. Forexample, at each marker, the profile tangents are in the plane tangentto the cylinder at the marker.

The user may then select and modify the markers, add additional markers,and/or modify the position and the direction of the isocline handlesand/or ribbon tangents. As the isocline ribbon is accordingly modified,the cylinder (more generally, geometric object) will reflect the changesin the modification of the isocline ribbon. Additional profiles andmarkers may be added in this manner until the desired shape of thegeometric object (derived from the cylinder) is obtained. An example ofthese steps is illustrated in the flowchart of FIG. 31. That is, theuser selects a graphically displayed surface (more generally, geometricobject) in step 1904. Subsequently, in step 1908, the user constructs aprofile curve on the selected surface (object).

Subsequently, in step 1912, an isocline ribbon (or at least the isoclineboundary) is generated for the profile. Note that this ribbon/boundarycan, if desired, be generated substantially without additional userinput. That is, an isocline ribbon/boundary may be generated from thetangency characteristics of the surface upon which the profile resides.In particular, for a parametric surface (more generally geometricobject), the parametric tangents on the surface at points on the profilecan be utilized to generate an isocline ribbon/boundary for the profile.Thus, surface neighborhoods on one side of the profile curve may be usedto determine a first isocline ribbon/boundary for a first surface havingthe profile, and if the profile is on the seam between the first surfaceand a second surface, then surface neighborhoods on the other side ofthe profile may be used to determine a second isocline ribbon/boundary.

Additionally, note that other surface characteristics may be preservedin an isocline ribbon/boundary. For example, in addition to preservingthe parametric tangents at profile curve points, the isoclineribbon/boundary may also optionally preserve the surface characteristicssuch as curvature, high order (>=2) derivative continuity with thesurface. Note, however, it is within the scope of the present disclosurethat further surface characteristics can be preserved in the isoclineribbon/boundary.

In step 1916, the generated isocline ribbon/boundary may be used tomodify the surface(s) having the profile curve as discussed hereinabovewith reference to the programs of the flowcharts of FIGS. 29 and 30.

In some embodiments of the user interface, an operation is provided tothe designer wherein a common boundary between two object space surfacescan be selected and the operation automatically forces the surfaces tojoin at a higher order continuity criteria (e.g., curvature continuity)than that of tangent plane continuity. For example, a higher ordercontinuity constraint imposed on an isocline ribbon deriving from one ofthe surfaces at the common boundary, can be used to similarly constrainan isocline ribbon for the other surface having the common boundary.Accordingly, this operation helps alleviate the so-called “Mach band”effect in which the human eye detects discontinuities in curvature insome instances.

Other user interface operations provided by the present disclosure are:

-   -   (a) Rounderizing, which is a tweaking operation that modifies an        existing surface to round off pointed edges, or to create darts        (i.e., surfaces that are smooth except at a single point, where        the surface kinks) that dissipate sharp edges. Such operations        can be performed using a computational system according to the        present disclosure by positioning profile curves on the surface        on opposite sides of a sharp edge and blending smoothly between        the profiles (e.g., using Formula (1) as described in Section        2.3.5; and subsequently eliminating the surface in between the        profile including the sharp edge.    -   (b) Embedding, which is an iterative user interface procedure        that can take one finished model, scale it, and perhaps rotate        or otherwise deform it to fit into part of another model.        4.1. Defining the Isocline via Markers, Profiles and the User        Interface

Explicit profiles are the profile curves that express a designer'sintent. Explicit profiles may be unconstrained (freeform) or partiallyconstrained (trim). Implicit profiles may be visible boundaries betweensurface patches caused, for example, by a surface discontinuity (i.e., akink or curve defined between an end surface of a cylinder and thecylindrical side thereof).

Implicit profiles are created automatically when the user introduces,e.g., a surface discontinuity. All profiles in a model are eitherexplicit or implicit.

4.1.1 Creating Markers

Profile markers and handles are created in the following ways:

-   -   A. Markers are automatically created at the ends of new implicit        and explicit profiles.    -   B. Inserted by a designer (e.g., by double clicking at a point)        on an explicit profile. To the designer, he/she is inserting a        point on the profile. The newly placed marker only minimally or        not at all changes the shape of the profile in the profile        segment containing the new marker. Subsequently, profile and        isocline handles are determined according to the shape of the        profile and surface(s) attached at the new marker.

A marker may be identified with a plurality of coincident points on thesame profile (e.g. a profile that loops back and attaches to itself).Such an identification of the marker with the plurality of profilepoints cannot be broken, except by deletion of the marker. In the caseof two or more profiles meeting at a common point having a profilemarker, such profiles each have a marker at the common point and themarkers are constrained to maintain coincidence so that moving onemarker will move both.

Profile markers inserted by the designer may be inserted for providingprofile handle points, or for setting specific isocline values. Notethat a profile handle point may have a set of constraints on itsisocline handles; i.e., isocline handle may inherit value(s) by aninterpolation of the nearest two adjacent isocline handles.

4.1.2. Viewing Markers and Profiles

Profile and isocline handles may have various constraints placed uponthem, wherein these handles may be displayed differently according tothe constraints placed upon them. In particular, the followingconstraints may be placed upon these handles:

-   -   (a) constrain a handle to a particular range of directions;    -   (b) constrain a handle to a particular range of magnitudes;    -   (c) constrain a handle to lie in a plane with other handles;    -   (d) constrain a handle to a particular range of curvatures;    -   (e) constrain a handle with a transform of another handle, e.g.,        identical rotations and/or translations.

The designer can choose to display the constraints through a displayrequest for the properties of geometric objects. In one embodiment,different colors may be displayed for the different types of constrainedprofile markers. For example, handles having no variability (alsodenoted herein as “fully constrained”) may be displayed in blue.

In some embodiments of the user interface, vectors are “grayed out” thatare constrained to thereby demonstrate to the designer that such vectorscannot be changed. For example, in one embodiment, fully constrainedhandles are typically grayed out.

4.13 Connecting Profiles Together

In one embodiment of the user interface, it supports the linkingtogether of two or more profiles that intersect at the same X, Y, Zlocation. Such intersection points are denoted “tie points,” when theparameterization of the point on each profile is invariant under profilemodifications. Note that such tie points may or may not have markersassociated therewith. When such a tie point is modified, allcorresponding profile curve points associated therewith at the tie pointare modified as a group. Such a tie point may be an endpoint of aprofile or an internal (i.e. “knot”) point.

Alternatively, a profile marker of a first profile may be constrained tolie within the object space range of a second profile (either implicitlyor explicitly). For example, referring to FIGS. 42A and 42B, the userinterface may provide the user with the capability to slide a profilemarker 2002 (contained on a first profile 2003 and a second profile2004) along the second profile 2004 for thereby changing the profile2004 of FIG. 42A into the profile 2004 of FIG. 42B when the marker 2002is slid along the profile in the direction of direction arrow 2006. Sucha slidable marker 2002 is denoted as a “slide point.”

Profile intersections are either “slide” or “tie” points. Moreover,these different types of points may be distinguished graphically fromone another by different colors and/or shapes. Note, if a profile slidesalong another profile, and the isocline ribbon for the sliding profileis used to compute a blended surface, S, then S will be recomputed.

4.1.4 Creating Markers and Profiles

The user interface may support the creation of a profile curve in anumber of ways:

A. Sketch the profile on the surface similarly to the data driventechnique of FIG. 28, wherein additional markers may be provided fortying the profile to the surface within a predetermined tolerance.Alternatively, in a second embodiment, a profile can be sketched acrossone or more surfaces by having the user select all markers for theprofile. Note that in either case, a profile may be sketched across oneor more surfaces. Moreover, in the second embodiment, the user interfacesupports the following steps to successfully create a profile.

(A1) Marker points are specified which lie on the surface(s). A fittedcurve is generated through the points in parameter space and then thisfitted curve is evaluated to obtain a corresponding image curve inobject space.

(A2) For each surface that the new profile crosses, the user may specifythe profile type (freeform or trim) for the portion of the profilecrossing the surface. A profile sketched on a surface either trims oneside of that surface or splits that surface into two new surfaces.Accordingly, note that any (non-degenerate) profile that enters theinterior of a surface must cross the surface's boundary at an enteringpoint and an exiting point. That is, the two surfaces along a commonprofile boundary are always linked to the profile, wherein, with respectto this profile, these surfaces may be characterized as follows: (a) onesurface is a trim surface and one is a non-trimmed surface (also denoteda “freeform surface”), or (b) two freeform surfaces.

B. Copy a Profile: A designer selects a profile to copy. The profile iscopied to a buffer (denoted a clipboard). The user then selects the modeof copy (examples: keep profile handles, or adapt profile handles to thegeometry where the profile is to be copied). The user selects where toposition the new profile (which may be additionally scaled, rotated,and/or mirrored, etc). The user selects a location for the new profile.Note that the new profile can be constrained by constraints on theoriginal profile. For example, the new profile may be mirrored about aplane from an existing profile, such that any change to the originalprofile will result in a change to the copy.

When a new profile is created, profile markers are automaticallygenerated at both ends of the new profile. Profile handles and isocline(ribbon tangent) handles are derived from the geometric characteristicsof the surfaces that the new profile splits.

4.1.5 Modifying Markers and Profiles

Modification of markers and/or (profile, isocline) handles is dependenton the constraints placed upon the markers and/or handles. They may beconstrained in one of the following ways:

A. Interactive modification by selecting a handle (profile or isocline)at a particular marker, m, and moving the handle endpoint so that it isconstrained to lie in a normal plane, i.e., either a plane normal to asurface having a profile containing m, or a plane defined by theisocline and profile handles of the profile at m. A pop-up propertysheet is also available for the user to key-in specific numerical valuesfor magnitude and angle for modifying a selected profile and/or isoclinehandle.

B. For markers that are constrained so that their pre-images lie withinthe pre-image of a profile, the marker may slide along such a parentprofile via, e.g., interactive modification by dragging such markerpoints. Note that for positional unconstrained markers, the markerpoints may be moved freely (i.e., under a user's direction and withoutother restrictions) along the parent profile(s) upon which such markersreside. Additionally, note that the user can select multiple profilemarker points by clicking on each, or by selecting all markers within adesignated region (e.g., bounding rectangle). Accordingly, the user isable to move a display pointing device (e.g., a mouse) which will thencause all of the selected markers to uniformly move in a correspondingdirection to the movement of the display pointing device. However,movement of the markers depends on the constraints set on these markers.For example, a constrained marker will only move within the limitsplaced upon it. Thus, if a first selected marker moves only within afirst profile and a second selected marker only moves within a differentsecond profile oriented orthogonally to the first profile, thendepending on the direction of movement desired, pone of the followingoccurs:

-   -   (i) the first marker is able to move, but the second marker can        not;    -   (ii) the second marker is able to move, but the first marker can        not;    -   (iii) both the first and second markers are able to move;    -   (iv) neither marker may be able to move.

C. Marker and handle constraints may be set by default rather thanexplicitly by the user. A pop-up property display form allows the userto set or remove specific constraints.

D. Additional constraints on profile and/or isocline handles may be setthat are dependent on the characteristics of other geometry. Forexample, profile and isocline handles can be constrained to be normal orparallel to a selected reference plane. Note that the position of aprofile marker also can be constrained to be dependent oncharacteristics of other geometry. For example, a marker can beconstrained to lie on a parting plane. That is, a plane of front/backsymmetry for designing a bottle. Another example of these constraints isin generating symmetric designs, i.e., a profile marker copy that isreflected about a parting plane will be constrained to be symmetric tothe parent profile marker.

E. Surfaces adjacent to a profile may have to satisfy either C0, C1, orC2 continuity, wherein C0 is positional, C1 is tangency, and C2 forcessmooth surface blends. One constraint that can be set on a marker is toforce C1 continuity between surfaces surrounding the marker bymaintaining equal length tangent vectors interior markers.

Freezing (eliminating the ability to modify) the profile and isoclinehandles at a marker will cause the profile segment containing the markerto rebuild based on the profile handles of the next two closest markers,effectively changing this marker's handles to reflect the curve built bythe two markers on either side.

For the most part, modifying profiles is a function of user interfacetechniques for modifying the profile marker positions and handles thatcontrol the shape of the profile. The following are examples of suchuser interface techniques.

A. Direct method: Profiles are modified directly on an object space (3D)model. This is done by modifying the profile markers and handles thatmake up the profile.

If the designer modifies a trim profile, the profile always lies withinthe parameter space of the surface it is trimming. That is, the trimprofile needs to be modified in the context of its original, overbuiltsurface within which it is embedded. Selecting a trim profile (or one ofits components) to modify causes the overbuilt, construction geometry tobe highlighted. It remains highlighted while the user is modifying thetrim profile.

A designer may have the option to turn on profiles and modify them usingthe direct method. For example, modifying a profile that defines anoverbuilt surface will cause the overbuilt surface to be updated. Sincethe profile that trims this overbuilt surface is constrained to liewithin the parameter space of the surface, the trim profile is alsorecomputed.

B. Design Ribbon method: This method is used to modify a specifiedregion of a profile. It allows, for example, the user to simplify thedesigner's interactions by modifying a profile in one view that iscomplex in another view. The designer identifies two markers that lie onthe same profile. The profile segment(s) between the two markers isextruded in at least one graphical view of the profile, thereby creatinga design ribbon (not to be confused with an isocline ribbon). The designribbon is a simple extruded surface (i.e., a curve which is swept alonggiven directions to generate a surface; for example, for markers at theends of a profile, offset the corresponding profile handles by theircorresponding isocline handles to obtain boundary handles andinterpolate an isocline boundary, e.g., by a lower dimensional versionof Formula (2); the surface having a perimeter consisting of theprofile, the isocline handles, and the isocline boundary defines the newextruded surface). The three-dimensional profile segments identifiedbetween the markers always lie within the pre-image of this designribbon. The user modifies the profile in one of two ways:

-   -   (B1) Modify the two-dimensional driving curve from which the        design ribbon was extruded, and which is instanced at the end of        the ribbon, wherein, by default, this curve is a two-dimensional        representation of the three-dimensional profile segments        defining the design ribbon. The user may “simplify” the driving        curve by selecting a subset of the two-dimensional points. Each        time the user modifies a driving curve point, the ribbon is        updated and the three-dimensional profile is modified to lie        within the parameter space of the modified ribbon. Operations on        the driving curve include any of those listed in the Profile        Marker section (point/slope modification, insert, delete, etc.).    -   (B2) Modify the two-dimensional profile points within the design        ribbon. The user directly modifies the two-dimensional profile        in a view that is perpendicular to the primary view. The        two-dimensional points always lie in the parameter space of the        ribbon. Operations include any of those listed in the Profile        Marker section (point/slope modification, insert, delete, etc.).

Only one design ribbon may exist per surface per profile segment. Designribbons may be created, modified, and deleted. Once they are created,they are persistent, i.e. they remain unmodified until a designermodifies the same segment at a later point in time. A design ribbon isdisplayed only when a designer is modifying it. A single profile mayhave multiple ribbons corresponding to multiple surfaces containing theprofile.

Note that modifying a profile using the direct method deletes any designribbons spanning the points being modified. This invalidates the ribbonand requires a designer to re-specify the ribbon.

C. Move Profile(s): The designer selects and moves two or more profilesin unison. That is, this user interface command selects all of theprofile markers on a profile (or segment thereof) and moves themtogether as a unit.

D. Merge Profiles: The designer may sketch a new profile and attach itto an existing profile so that an endpoint of each profile iscoincident. Additionally, the designer may specify which segment orsegments of the existing profile to delete. Subsequently, the newprofile and the remaining connected portion of the existing profilehaving the coincident end point with the new profile are merged. Notethat merging profiles causes the set of the respective profile handles,isocline handles and ribbon tangents for each of the two coincidentendpoints to be combined into a single such set.

E. Split profile: Split one profile into two at a single point, p. Anendpoint of each of the two new profiles is constrained to be coincidentat p.

4.1.6 Deleting Markers and Profiles

Deleting a profile marker is always possible, except at the endpoints ofa profile. However, some embodiments of the present disclosure may needto replace a marker with a marker having constraints if it is needed formaintaining smooth patches. Note that a new constrained replacementmarker may or may not be in the same location as the previous marker.

If an entire profile is deleted, then the user interface both highlightsany dependent geometric object(s) and requests user confirmation beforedeleting the profile and the dependent geometric object(s). Accordingly,note that a computational system according to the present disclosureretains sufficient dependency information regarding dependencies betweengeometric objects in a model so that for modifications of an object thatis used for deriving other objects, appropriate additional modificationscan be performed on these other objects automatically.

4.1.7 Profile Markers and Handles

Note that there are typically two profile handles, two isocline handlesand two ribbon tangents for a profile marker, i.e., a profile handle, anisocline handle and ribbon tangent per surface on each surface havingthe profile as a boundary curve. However, there may be more handlesassociated with a profile where several profiles converge or fewer ifthe profile is the edge of a surface.

4.2 Isoclines and the User Interface

The slope of an isocline handle controls surface tangency at a markerand at a surrounding portion of the profile containing the marker. Themagnitude of an isocline handle controls the fullness of the dependentsurface. That is, how much the surface bellies out. An isocline handlemay be constrained to be offset from another isocline handle (i.e. −10degrees from other side). An isocline handle can be calculated at anypoint along a profile (by inserting a marker on the profile).

4.2.1 Creating an Isocline Handle

The user interface supports the constraining of isocline handlesrelative to one another. Such handles can be forced to always betangent, of equal magnitude, or offset by some amount. In one embodimentof the present disclosure, the user interface provides a pop-up menu todisplay and change isocline handle constraint values, such as length anddirection.

4.2.2 Modifying an Isocline Handle

If the user slides a profile marker along a profile, the user may fixthe isocline handle for the marker, thereby causing the surfacesadjacent to (and dependent upon) the profile to change or have theisocline handles interpolated between the nearest two isocline handleson the profile (which case implies that the dependent surfaces are notaffected).

4.2.3 Deleting an Isocline Handle

The user interface supports the deletion of isocline handles.

4.3 Special Geometric Objects and the User Interface

The present disclosure provides for the creation and manipulation of anumber of specialized geometric object types that can substantiallyfacilitate the design of objects such as containers.

4.3.1 Label Surfaces

A label surface is a special case of a trimmed surface. The special caseaspects of a label surface are:

-   (i) there is a “watershed” profile that runs from the bottom to the    top of the label;-   (ii) there are label curves between which the corresponding label    surface is ruled (e.g., label curves 132 of FIG. 11);-   (iii) there is a boundary (trim) profile (e.g., trim profile 138 of    FIG. 11).

The key difference that makes a label surface different from othertrimmed surfaces is that the original surface (from which the labelsurface is trimmed) is a ruled surface. In particular, the label surfacedefining curves are constrained such that a ruled surface is maintainedwithin the boundary of these defining curves.

Note that other surfaces may be blended to a trim profile for a label,but the trim profile can only be modified in a manner that insures thatit bounds a ruled surface.

In one embodiment of the present disclosure, a two dimensional “rolledout” representation of the label surface can be generated. That is, thesurface can be associated in a one-to-one fashion with the plane byrolling it flat. Such a representation simulates a label surface inwhich a designer can thereon create a piece of artwork that cansubsequently be wrapped on a container.

4.3.1.1 Creation of a Label Surface

To create a label surface, an overbuilt surface to be trimmed must be aruled, approximately developable surface, i.e., a ruled surface in whichall surface normals on any ruling are parallel. Subsequently, the userthen follows the normal trim surface steps; i.e., sketching a profile onthe ruled surface, generating the (sub)surfaces on both sides of theprofile (i.e., the label surface and the portion of the surface to betrimmed away), trimming the label surface and blending other surfaces tothe trim profile.

Note that the trimmed away surface portion is hidden from normal viewing(i.e., it is no longer a part of the visible model).

The following procedure may be provided for generating a label surface.At a high level, the steps for this procedure are:

-   (i) Make sure the surface is ruled. That is, the user interface    supports automatic modification of user selected profiles so that    these profiles satisfy 4.3.1(i) and (ii). In particular, to perform    this step, the following substeps are performed:-   (ii) The user sketches a boundary profile on the surface defining    the bounds of the label;-   (iii) Construct a graphical representation of a label (i.e., a ruled    surface having text, artwork, and/or designs thereon);-   (iv) Allow the user to graphically apply the label representation to    the label surface (or a representation thereof). In particular, the    user interface for applying the label representation may    automatically attach the label representation to the label via a    grouping type of operation so that the label representation    maintains its position on the label surface during, e.g., label    surface rotations, translations, scaling operations, etc.; and-   (v) Allow the user to undo the design when the label surface and/or    the label is not satisfactory.    4.3.1.2. Modification of a Label Surface

Modification of the label surface components is somewhat different thanthose of a trimmed surface.

The portion of the watershed profile that is a straight line segment isconstrained to remain straight.

The boundary opposite of the watershed (the “other side”, of the parentruled surface) cannot be modified. It is simply a straight line segmentbetween the top and bottom boundaries. The top and bottom boundaryprofiles can be modified. They are constrained so no additional freeprofile markers can be inserted on them. Also, the profile marker at theend away from the watershed is constrained to move only to maintain aruled surface. It can be extended (extrapolated along the samecurvature) and the angles at its endpoints can be adjusted—again, aslong as it maintains a ruled surface.

4.3.1.3. Deleting of a Label Surface

Deleting a label surface removes the constraints on all of the profilesused in creating the label surface. Additionally, all of theconstruction geometry for the label surface that is invisible to theuser will also be deleted. The constraints for maintaining a ruledsurface will also be removed. Thus, the remaining geometric objects arethen freed from the label surface constraints, and can be modified inways not previously available.

4.4. Hole Tool User Interface

The present disclosure also provides a user with a novel computationalmethod that helps the user add a hole to a geometric model (such as foradding a handle to grasp to a non-handled bottle). The informationrequired to add a hole to a model using this procedure includes: a loopof profile segments on a front surface, a loop of profile segments on anopposite back surface, the type of each profile in a loop of profiles(freeform or trim), and optional profile(s) to shape the interior of thehole.

4.4.1. Creation of a Hole

The hole creation tool guides the user through a series of steps to adda hole. FIGS. 21A-21C illustrate the procedure for creating a hole 600(FIG. 21C) on a geometric object 604 using a computational systemaccording to the present disclosure. The corresponding steps performedfor creating the hole 600 are as follows:

-   (a) Sketch a profile loop 608 on the front surface 612;-   (b) Sketch a profile loop 616 on the back surface (optionally    project the profile 608 to the back surface).    Note that isocline handles are automatically placed on both profiles    608 and 616.-   (c) If one or more of the profiles for one of the profile loops 608    and 616 are freeform profiles, then the user may add new profiles    (to complete such a profile loop), and/or profiles merge    corresponding to such a loop whereby these profiles are constrained    so that they are utilized as if they were a single profile.    Accordingly, once the profile loops are constructed, then surfaces    may be skinned between the loops to thereby replace the original    surface occupying the hole.

If a trim profile loop is specified, the surface region inside theprofile loop is trimmed.

If specified, the hole creation procedure uses additional profiles toplace and shape surfaces on interior boundaries for the hole. Otherwise,surfaces are skinned automatically between the front and back profileloops.

4.4.2. Modification of a Hole

Modifying a hole is a function of modifying profile markers and handlesthat make up the geometry of the hole.

4.4.3. Deleting of a Hole

Deleting a hole is also a function of deleting the components that makeup the geometry of the hole; i.e., profiles and other geometry for thehole.

4.5. Smoothness Considerations via the User Interface

We consider the order of transition between adjacent surfaces (whichmeet at the profiles). This section does, however, contain some broaderimplications for the general theory in Section 2.

4.5.1. Continuous Profiles via the User Interface

Given two profile curves that intersect, derivative continuity across anintersection point may be assured if several conditions are met:

-   (a) an end point of one profile is coincident with an end point of    the other profile (positional continuity);-   (b) the blending functions B_(i) used in generating the profiles (as    per FIGS. 26 and 27) are equal at the intersection marker, and-   (c) the profile handles at the intersection marker are collinear and    equal length.

Tangent directional continuity is a weaker condition that can besatisfied if condition (c) above is changed to:

(c*) the profile handles at the intersection marker are only colinear.The magnitudes of the profile handles may differ in this case.

The designer may intentionally produce a kink at a marker by breakingthe collinearity of the two profile handles at the intersection marker.This means that the two profile handles do not have a common direction.

4.5.2. Continuous Surfaces

The notion of tangent plane continuity between surfaces may be definedas follows: for each point p of a boundary between two surfaces S₁ andS₂, the tangent plane, T₁(p), of S₁ at p is identical to the tangentplane, T₂(p), of S₂ at p. To achieve tangent continuity between surfacesacross a profile boundary therebetween, it is necessary that theisocline handles (for each of the surfaces) at each marker on theprofile boundary lie in a common plane with each other and the profilehandle at that marker. If this is not done then a kink in the surfacealong the profile will be created.

Note that when there are two or more surface patches to be generatedwherein these patches must be constrained to meet at a common markerpoint, p, a computational system according to the present disclosure mayautomatically generate isocline handles, denoted “common directionhandles.” That is, for each profile, P (having p) used in defining oneor more of the surfaces, there may be a corresponding automaticallygenerated common direction handle which is a vector, V, oriented from p,wherein V is perpendicular to the profile handle of the profile P, thisprofile handle lying in the common plane formed by the profile handlesfor the other profiles also having the point p. Further note, the userinterface supports allowing the user to either display or not displaythe common direction handles.

Note that it is not necessary to have profile handles and isoclinehandles that match for profiles that adjoin at a common marker in orderto achieve smooth surfaces there, only that they all lie in a commonplane. In FIG. 18, there are three profile curves 404, 408 and 412 forthe surfaces 416 and 418. Each of the three profile curves meets at theprofile marker 420, and each of the profiles has a correspondingisocline ribbon 424 (for profile 404), 428 (for profile 408), and 432(for profile 412). Additionally, the profile and isocline handlesassociated with the profiles 404, 408 and 412 and the marker 420 are:

(i) profile handle 436 and isocline handle 440 for profile 404;

(ii) profile handle 444 and isocline handle 448 for profile 408;

(iii) profile handle 452 and isocline handle 456 for profile 412.

Thus, if the profile and isocline handles 436, 440, 444, 448, 452 and456 all lie within the plane 460 (indicated by the dashed rectangularportion), then the surfaces 416 and 418 smoothly join at the marker 420.

At any marker, two isocline ribbons are likely to meet in the way thattwo profiles may meet, that is, two isocline ribbons may have a commonisocline handle as an edge for each of the ribbons.

To achieve tangent plane continuity between different (blended) surfaceregions S₁ and S₂ (FIG. 43) joined by a composite profile (having theprofiles P₁ and P₂ therein), not only is tangent continuity acrossprofiles P₁ and P₂ needed, but tangent continuity between adjacentribbons R₁ and R₂ is also needed. That is, for the profiles P₁ (betweenmarkers 2010 and 2014) and P₂ (between markers 2014 and 2018), therespective ribbons R₁ and R₂, when thought of as surfaces, must betangent plane continuous, and share a common isocline handle 2022. Notethat in most cases, tangent continuity between ribbons is equivalent totangent continuity between profiles and tangent continuity betweenribbon boundaries that is required for smooth transitions across surfacepatch boundaries. Moreover, the user interface of a computational systemaccording to the present disclosure provides techniques for assuringtangent plane continuity between ribbon boundaries wherein thesetechniques are substantially identical to those used for assuringtangent plane continuity between profiles. Thus, the present inventioncan provide tangent plane continuity between adjacent surfaces generatedfrom isocline ribbons according to the present disclosure.

In some circumstances, it is possible to break the continuity ofcomposite ribbons intentionally, thereby causing a crease across thesurface generated from the ribbons wherein the crease does notcorrespond with a coincident profile along the crease. However, in someembodiments of the present disclosure, an “implicit profile” can becreated that is coincident with the crease.4.5.3. Curvature Continuity

The visual quality of a surface depends not only on tangent planecontinuity, but also on higher order derivatives. A user can be acutelysensitive to discontinuous changes in surface curvature, especially ifthe surface is rendered with specular highlights or reflected texturemappings, which is common in simulating realistic scenes. The user mayperceive a distracting visual artifact known as a “Mach band.”Accordingly, raising the order of continuity between transitions to thatof curvature continuity ameliorates this.

Analysis has shown that the curvature of the surface defined by Formula(1) or Formula (4) depends on the second derivatives of the B₁ and theS_(i). The dependencies of the B_(i) are non-trivial and it isadvantageous to choose the blending functions so that their secondderivatives are zero and let the surface functions S_(i) determine thecurvature. The cosine squared function of Section 2.1 fulfills thiscondition. There also exist certain quintic polynomials that aresatisfactory. For example, the polynomial B₁(x) of Formulas (3).

If the curvature of a blended surface generated from Formula (1) orFormula (4) depends only on the S_(i) (e.g., B_(i)″=0), it is thenpossible to raise the curvature order between the bounding surfacepatches S_(i) analogous to the methods in the previous section forachieving tangent continuity. To do this, simply define thecorresponding profiles and isocline handles so they match in theirsecond derivative at each marker along the profile boundary. Note,however, that each profile handle may be considered as a linear functionof one parameter and therefore has a zero second derivative. Thuscurvature continuity is achieved; albeit by making the curvature acrossthe profile “flat,” i.e., zero. This is useful at points where there isan inflection point on the profile, but can be undesirable elsewhere. Torectify this situation, the linear handles may be replaced with curvedribs, such as parabolic arcs. Accordingly, the handles now become arcs,and at the markers, the curvature is made to match that of the givenarc. By extending the concept of providing a nonzero curvature to allhandles, e.g., profile, isocline and boundary handles, along with thezero second derivatives of the blending functions and the effects of theMach banding can be mollified.

4.5.4. G1 Continuity Using Roll, Yaw and Magnitude

The present disclosure also provides a user interface method to specifyhandle vectors (e.g., isocline handles) relative to a correspondingprofile curve, wherein G1 continuity (as defined in the Definitionsherein above) between surfaces joined together by the profile isassured. This method, which is denoted herein as the roll-yaw method,specifies a vector V in terms of three scalar terms called roll, yaw andmag (magnitude), wherein roll and yaw are determined at a point P on acurve using the tangent vector T at the point P, and a vector N normalto the curve at the point P. The yaw component of the vector Vrepresents the angular deviation from T at P. For instance, if thevector V is in a direction perpendicular to T, the yaw value (in atleast one framework) is 0°, and if the vector V at P is in the samedirection as T, then the yaw value is 90°. Regarding the roll componentof the vector V, this scalar represents the amount of angular rotationabout T as the axis of rotation, and wherein the baseline axis formeasuring the angle is the vector N at P. Accordingly, the vector Nrepresents 0° of roll and the rotational range extends from −180° to180° using the right-hand rule, as one skilled in the art willunderstand. Regarding the magnitude component of vector V, this issimply the length of the vector V. Note that any vector expressed interms of three-dimensional Cartesian coordinates can be transformedone-for-one into the roll, yaw, mag notation for a given T and N.

Note that the vector N may be selected from among vectors in the planenormal to T. However, this does not precisely define N. Thus, severalmethods may be used to define N. A first such method for defining N issimply to choose a constant vector VC and then determine N by thefollowing equation: N=T×VC. This method, however, produces an undefinedvalue for N when T and VC are colinear. To provide appropriate valuesfor N where this equation yields a zero vector, N can be approximated ina topological neighborhood of the colinearity. Alternatively, in asecond method of generating N, the Frenet-Serrat frame of the underlyingcurve may be chosen, as one skilled in the art will understand. However,the Frenet-Serrat frame may be discontinuous at inflection points alongthe curve. Accordingly, the present disclosure provides a method forcreating a minimally rotating reference frame for a complex (i.e.,three-dimensional) curve that obviates difficulties in defining thevector N regardless of the orientation or shape of the curve and itstangent vector T.

As previously mentioned the roll-yaw method provides a novel way toachieve G1 continuity across a profile. As an example, consider thegeometry illustrated in FIG. 44, wherein a profile P along with left andright isocline ribbons LR and RR are shown. Each of the isocline ribbonsLR and RR has two corresponding isocline handles at its ends, i.e., HL1and HL2 for LR, and HR1 and HR2 for RR. Assuming the profile P endpointshave handles denoted HP1 and HP2, for any point pp on the profile,continuity across the profile for surfaces bounded thereby is determinedby the interpolated isocline values IL and IR. Further, IR isinterpolated (according to the techniques of the present disclosure)from HR1 and HR2, and IL is interpolated from HL1 and HL2. Thus, oneskilled in the art will understand that for G1 continuity across theprofile P, IL and IR must at least be in opposite (colinear) directions.Further, it can be shown by one skilled in the art that if IR and IL areformed using a cubic Hermite interpolation between HL1 and HL2 for IL,and, HR1 and HR2 for IR that the conditions for G1 continuity are thatHL1 and HR1 must be equal and opposite vectors. Further, the same mustbe true for HL2 and HR2. However, if instead of interpolating theisocline values IL and IR in Cartesian space, the interpolation isperformed in (roll, yaw, mag) space, G1 continuity can be achieved witha less strict condition, namely, that the roll value of HL1 and HL2 mustbe the same. Accordingly, this is equivalent to saying that HL1, HR1 andHP1 must be no more than coplanar (with the same being true for HL2, HR2and HP2) in order to guarantee G1 continuity everywhere on the profileP. Further, note that similar conditions may be imposed if the isoclinehandles are curved rather than straight. In particular, tangent vectorsto the isocline handles at their common points with the profile P may beused in place of any corresponding isocline handle vector represented inFIG. 44. Thus, as one skilled in the art will appreciate, computationalsteps can be provided that embody the roll-yaw method for, if necessary,converting from Cartesian vectors to roll, yaw, mag vectors, and thenassuring that the above described coplanar constraint is satisfied forguaranteeing that surfaces are G1 continuous across the profile P.

4.6. Embedding Models within Models

The present disclosure allows parts of a surface bounded by profiles tobe designed separately from one another. For example, a triangularportion of a surface may be designed as a free standing surface model.That is, a designer may add profiles and isocline ribbons as desireduntil a satisfactory design of the model is obtained (using barycentricmappings as one skilled in the art will understand). Afterwards thispiece may be distorted, rotated and fit into a triangular portion ofanother model. Hence, a finely detailed model may be designed andembedded into another model. By maintaining links this process can beused for level of detail management. That is, for example, when themodel is viewed from a distance, the detailed portion is unneeded fordisplay, but as the viewer moves closer the embedded object is linked infor the extra detail it affords. Two examples of types of embeddingsfollow in the next subsections.

4.6.1. A Rounderizing Technique

Referring to FIG. 19, a small blended surface rounds an edge 482 betweentwo intersecting surfaces 484 and 486. This blended surface 480 isblended from the thin surface strips 488 and 490 whose pre-images are a“small” offset from the pre-image of the edge 482 in parameter space.This process is a straightforward application of Formula (1) where thetwo surfaces 484 and 486 are blended using their common parameter space(not shown).

The new surface types lead to new evaluation routines that areespecially efficient in special cases described.

5. Evaluation

We will first consider the evaluation of the two-edge blend, recognizingthat other forms derive from this fundamental form. Because of itsimportance we will recall Formula (1), which isS(u,v)=S ₁(u,v)B ₁(u,v)+S ₂(u,v)B ₂(u,v).  (1)There are both blending functions B_(i) and isocline ribbons S_(i) todetermine when evaluating the surface S. The blending function iscalculated as a univariate function of distance in the parameter space.As discussed in Section 2, the evaluation of the distance functionvaries considerably depending on how complex the pre-image is inparameter space. Once determined, the actual blending value can becalculated by a simple table look up; that is, the blending functionsare tabulated to a sufficiently high resolution and stored in memorywhere they can be indexed by the input variable. Consider the functionof B₁(x) of Formula (3). Evaluate this function at x=0, 0.01, 0.02, . .. , 0.99, and 1. These 1001 values are stored as an array. When a pointX is given, it is used to locate the nearest point in the array, e.g.,between 0.52 and 0.53. Subsequently, B(0.52) or B(0.53) are used as thefunction value.

There are many techniques that may apply based on what the distance andisocline ribbon functions are. The present discussion is focused on amethod that assumes a simple model computationally, but nevertheless,retains considerable design flexibility. The isocline ribbons 508 (S₁)and 516 (S₂) will be given as in FIG. 20. These are parameterized from 0to 1 in both u and v parameters. For each fixed value of v along theprofile line 504, if the corresponding picket on isocline ribbon 508 isa straight line segment (e.g., line segment 512), the isocline ribbon isa ruled surface as one skilled in the art will understand. Accordingly,the parameter u provides a distance-like measurement along the rulingwhere the point (u,v) is found. Assume that each of the isocline ribbons508 and 516 are ruled surfaces. Further assume that the pre-image ofeach of the profiles 504 and 506 in parameter space are the profilesthemselves and the distance-like measurement is the parametric u valueof a point (u, v₀) on the v₀ ruling of the isocline. Because theisocline ribbons 508 and 516 are ruled surfaces, for the constant v₀parameter we can scan out a set of equidistant points along linesegments 512 and 520 by simply adding the appropriate offset vector tothe previous value. The initial value is S_(i)(0, v₀). The offset vectoris obtained asT ₀=[S ₁(1,v ₀)−S _(i)(0,v ₀)]/n,  (10)where n is the number of points desired on the ruling line to scan fromone isocline ribbon (pre-image) edge to the opposing other edge.

If we impose the restriction that the blending functions are a partitionof unity, i.e., B₁=1−B₂, which is desirable from a design perspective,then the Formula (1) yieldsS(u,v)=[S ₁(u,v)−S ₂(u,v)]·B ₁(u,v)+S ₂(u,v)  (11)

In one embodiment, this form and with the previous simplifications, itis seen that each point requires three vector adds (for S₁, S₂ and the“+”), one table look up (for B₁(u,v)) and one scalar multiply. This isafter initialization which consists of finding each S_(i)(0, v) andcomputing T₀, the offset vector (using Formula (10)). To scan out a setof points on S, one simply increments through the parameter v, and thencomputes points along the rulings in u.

In the case of the defined four-edge surface (as in Section 2.2), someS_(i) are as the two edge case above, but the others blendlongitudinally across the ribbon first. Specifically, in FIG. 34 thev-loft case is the same as FIG. 35 with re-labeling, while the u-loft ofFIG. 34 is a horizontal blend of isocline ribbons. The four-edge surfaceresults from the barycentric blend of all four.

In FIG. 33, four profile curves P₁₁, P₁₂, P₂₁ and P₂₂ are shown betweenwhich a surface is desired to be placed. In FIG. 34, the profiles P₁₁and P₁₂ together with their corresponding respective isocline ribbonsR₁₁ and R₁₂, are used to create the blended surface S₁.

While S₁ evaluates exactly as the two-edge case already described, thegeneration of S₂ (FIG. 35) differs because the u and v parameters arereversed. In this case, the straight line segment on the isoclines R₂₁and R₂₂ corresponds to fixing u and scanning in v; a situation which isincompatible to rapid scanning. However, it is desirable to fix just oneparameter and scan the other in both FIGS. 34 and 35. In one embodiment,this can be resolved by defining the isoclines R₂₁ and R₂₂ differently.That is, each such ribbon becomes a blend of two ruled surfaces definedby simple user inputs. For example, consider isocline ribbon R₂₁. It maybe defined by blending two bilinear surfaces 1950 and 1952 in a mannersimilar to the surface generation techniques described in section 2.2and illustrated in FIGS. 37 through 39. That is, the edges of thebilinear surfaces 1950 and 1952, that are tangent on the profile P₂₁,are the profile handles 1956 and 1960; the boundary handles 1964 and1968 are tangent to the ribbon boundary 1972 and form the opposite edgesof the bilinear surfaces. The other two line segments 1976 and 1980 areuser inputs.

It is now possible to fix v in the second (u-loft) as well and scan byadding a single vector offset. This effort produces points on theisocline ribbons, R₂₁ and R₂₂, each at the identical cost of producingpoints on the v-loft. Additionally, we must blend the new points tocompute the point on S₂. In operation counts, there are, therefore,eleven vector additions, five scalar multiplies and one table look-up.The additions include three for the v-loft, three each for the u-loftisoclines, one for blending these isoclines and one for blending the twolofts.

For the general N-sided surface it is first necessary to compute adistance on each ribbon. The parameters are calculated using the N-sidedparameterization technique from Section 2.2. These distances are thenplugged into the blending functions of Formula (6). They are adjusted sothey vary from 0 to 1.

The parameters for the ribbons must be set from the distance so given.That is, one parameter will be the distance (from the profile). Theother parameter can be deduced by determining where the parameter lineof FIG. 12 crosses the edge of the N-sided polygon. It is assumed thatthe polygon has edges of length 1. After these parameters are figuredthen Formula (4) has all constituents needed for calculation.

6. Conversion of N-Sided Object Space Surfaces to Standard NURBS Form

Referring to FIGS. 47-50, these figures show the same image surface 3020generated by a novel N-sided surface generation technique, wherein thegreen 3-sided patch 24 (FIG. 47) is of polynomial degree 8, while thered 4-sided patch 3028 (FIG. 47) is of polynomial degree 10; both ofrespectably low degree. In particular, it is believed that surfacesgenerated by polynomials of degrees such as these do not posedifficulties for engineering applications such as computer aidedpart/model design (e.g., of auto bodies, bottle shape, prosthetics,aircraft frames, ship hulls, animation objects such as faces, etc.).Note that the 4-sided patch 3028 has a sharp corner at the lower rightvertex (FIG. 47). Of particular interest is where the patches abut withidentical tangency. Moreover, the surface 3020 is visually watertight.Moreover, note that an embodiment of this novel N-sided surfacegeneration technique is provided in Appendix A hereinbelow which showsMATLAB code for implementing this technique.

Referring to FIGS. 49A and 49B, these figures show the smoothness acrossthe boundary between the two patches 3024 and 3028 of FIG. 47. Inparticular, isophote stripes are displayed on the surface, wherein theisophote stripes extend across the boundary joining both patches 3024and 3028. As can be seen, the patches appear to smoothly deform into oneanother across their mutual boundary.

Note that isophotes paint all parts of the surface that have an equalangle to the direction of a spot light with the same color. They arevery sensitive to discontinuities and unwanted variations in thesurface. Auto stylists use them to help shape surfaces with goodreflection properties. The sharp point at the lower right vertex isapparent with this method of rendering. Notice especially the smoothtransition between the patches.

FIG. 50 renders the patches 3024 and 3028 with alternating red and greentriangular facets, wherein the variation in the parameterizations acrossthe surfaces of both of the 3-sided patch 3024 and the 4-sided patch3028 are illustrated. In particular, each of the red and green trianglescorrespond to an area extent in the parameter space that is of identicalsize and shape, and, the smoothly deforming contour curves defined bythe triangles are the images of uniformly spaced parallel lines in theparameter space.

6.1 Low Degree Patch Description

To make exposition of the novel surface generation technique simpler anillustrative surface S(u,v) is defined hereinbelow that interpolates toobject space boundary curves, i.e., the object space surface S(u,v)defined by Formula (12) below has interior points that are smoothlyinterpolated from the surface's boundary curves, and the image (inobject space) of a polygon in the domain parameter space for the surfaceS(u,v) matches exactly these boundary curves. Note, such a surface maybe generated according to the disclosure in U.S. Pat. No. 7,236,167incorporated herein by reference which describes the surface generatingFormula of (12) below.

Using Formula (12) below, a technique for generating N-sided patches canbe provided. Moreover, by matching patch tangency constraints, a G¹composite surface can be generated.

Referring to Formula (12) below, for points (u,v) of a parameter spaceobject (e.g., a polygon), the corresponding object space surface S(u,v)may be defined:S(u,v)=Σ_(i=1,N)[ƒ_(i)(t _(i)(u,v))Π_(j=1,N) ,W(d _(j)(u,v))  (12)where the functions d_(j)(u,v) are distance-like functions in parameterspace from the j^(th) parameter space polygonal edge as described inU.S. Pat. No. 7,236,167, and wherein a possible additional weightfunction

(x) may be:

(x)=x ²,  (13)and a function for each of the boundary curves ƒ_(i) through which thesurface S(u,v) must pass (each such curve also referred to as a “featurecurve”) is:ƒ_(i)(t)=RW(t)L _(i)(t)+(1−RW(t))R _(i)(t)  (14)for the weighting function RW that is smooth and satisfies theconditions that:RW(0)=1, RW(1)=0, RW′(0)=0 and RW′(1)=0.An example of such a function is:RW(t)=(1−t)²/(2t ²−2t+1).  (15)Also in (3), L_(i) and R_(i) may be defined as the following linearinterpolantsL _(i)(t)=(1−t)p _(i,0) +tp _(i,1) where 0≤t≤1  (16)andR _(i)(t)=(1−t)p _(i,3) +tp _(i,2) where 0≤t≤1  (17)with points {p_(i,0), p_(i,1), p_(i,2), p_(i,3)} for defining the i^(th)boundary curve ƒ_(i) of a patch (in object space) as in FIG. 51, whereinthe i^(th) boundary curve and these points are the end points of, e.g.,profile handles as disclosed in U.S. Pat. No. 7,236,167. Note that bymaking such boundary curves ƒ_(i) sufficiently small in length, anyarbitrary curve in object space can be approximated to a predeterminedtolerance as described in U.S. Pat. No. 7,236,167.

Furthermore, the parameter functions t_(i)(u,v) and d_(i)(u,v) ofFormula (12) above, are each derived from the i^(th) edge of a basepolygon in (u,v) parameter space as shown in FIG. 52. Note that as shownin FIG. 52, the base polygon may be a triangle. However, such a basepolygon may be a quadrilateral for the 4-sided case, a pentagon for the5-sided case, and so forth.

The “t-value” function of (1) may be defined as followst _(i)(u,v)=((s _(x) −u)(s _(x) −e _(x))+(s _(y) −v)(s _(y) −e_(y)))/((e _(x) −s _(x)+(e _(y) −s _(y))²)  (18)

Note that Formula (18) computes t_(i)(u,v) to be the parameterizationvalue (between 0 and 1) for a point p(u,v) (FIG. 52) on the i^(th) sideof the base polygon in parameter space, wherein p(u,v) is theperpendicular drop point or projection of (u,v) onto the side as isshown in FIG. 52. Moreover, t_(i) is linear in u and v.

Various alternative computations for t_(i)(u,v) are also within thescope of the present disclosure. In particular, referring again to FIG.52, t_(i)(u,v) may be alternatively defined as follows:t _(i)(u,v)=|(m+b)−v|, or, t _(i)(u,v)=|(v−b)/m−u|.

The “distance” function d_(i)(u,v) may be defined as follows:d _(i)(u,v)=(mv+b−u)sin(arctan(m)).  (19)Note that d_(i)(u,v) may be considered a distance from (u,v) to the linev=mu+b (FIG. 52) with slope m and v-intercept b. Further note that ifthe slope m is vertical, then the distance d_(i) is defined to be b−u.However, various alternative computations for d_(i)(u,v) are within thescope of the present disclosure. In particular, d_(i)(u,v)=[(mv+b−u)sin(arctan(m))]² for m not vertical, and (b−u)² otherwise. However, dueto the increased polynomial degree of this alternative embodiment,Formula (19) is preferable. Note that Formula (19) is linear in u and v(low degree) since the sine term is constant for the entire edge.

Although various weight functions (

) and distance functions (d_(i)) may be used for the composition

(d_(i)(u,v) in Formula (12) above, each such composition function mustbe 1 when (u,v) is on the i^(th) side of the base polygon, since thiscondition implies that (1−W(d_(i)(u,v))) is zero, and accordingly theinfluence of the other boundary curves is nullified thereby enablingsmooth interpolation between surface patches, as one skilled in the artwill understand.

The weight function W in Formula (13) may be replaced by RW in Formula(15). Such replacement raises the denominator degree, but not the maxdegree, which is in the numerator (see below).

The reason for preferring Formula (14) for defining curves over astandard form, such as a Bezier form, is that Formula (14) reproducesexact extrusions of its curves. This is due to the fact that Formula(14) is the curve form of Formula (12) if RW is used instead of Wdefined in Formula (13). However, Bezier, or any other parametric formcould be used as one of skill in the art will understand. Note, the formof RW in Formula (15) was devised to satisfy RW(0)=1, RW(1)=0, andRW′(0)=RW′(1)=0, while being of low degree.

Most distance forms use a square and square root. However, Formula (19)is linear in u and v. The factor sin(arctan(m)) is constant for u and v,and may be only computed once per base polygon edge, thus additionallyincreasing computational efficiency.

In at least some preferred embodiments, the base polygons for each patchshould be determined so that their sides roughly correspond in length tothe lengths of the boundary curves.

6.2 Reckoning the Surface Degrees

Starting at the interior of the surface S(u,v) of Formula (12), it canbe seen that both the t-value and distance of Formulas (18) and (19),respectively, are linear in u and v; thus each are of degree 1. InFormula (12), each instance of the feature curve function ƒ_(i)(t_(i))of Formula (14) is a sum of two products, where each product is aquadratic function (according to Formula (15)) times one of the linearfunctions of Formulas (16) or (17). Thus, ƒ_(i)(t_(i)) has polynomialdegree 3 over degree 2 in parameter t (and accordingly, also in u and vsince t may be linear in u and v).

Composing the t-value function t_(i)(u,v), which as noted above isdegree 1, into Formula (14) does not raise the degree of the resultingfunction ƒ_(i)(t_(i)(u,v)). Each product of the weight functionsΠ_(j=1,N,i·j)

(d_(j)(u,v))(1−W(d_(i)(u,v)) in Formula (12) is multiplied by acorresponding feature curve function, f_(i)(t_(i)(u,v)). Since thelinear distance functions d_(i)(u,v) adds nothing to the degree of theweights W and (1−W), the weight functions W(d_(i)(u,v) and(1−W(d_(i)(u,v)) are each quadratic, i.e., of degree 2. In the casewhere the parameter space polygon is a triangle, each termΠ_(j−1,N,i≠j)W(d_(j)(u,v)) of Formula (12) is the product of two weightfactor pairs W(d_(j)(u,v)) each of order 4 (in u and v), therebyyielding a polynomial of degree 8. For the rectangular case, each termΠ_(j=1,N,i≠j)W(d_(j)(u,v))(1−W(d_(i)(u,v)) of Formula (12) is theproduct of three weight factor pairs W(d_(j)(u,v))(1−W(d_(i)(u,v)), eachof order 4, thereby yielding a polynomial of degree 12. Then we see thatthe total degree is obtained by the multiplication of the feature curveterm f_(i)(t_(i)) with the weight termΠ_(j=1,N,i≠j)W(d_(j)(u,v))(1−W(d_(i)(u,v)). E.g., a degree 3 over 2 termmultiplied by a degree 8 term (for the triangular case), and, a degree 3over 2 term multiplied by a degree 12 term (for the rectangular case).Multiplying terms adds degrees, so the total degree of Formula (12) willbe degree 11 over degree 2 for 3-sided patches and degree 15 over degree2 for 4-sided patches. Moreover, for a 5-sided polygon (instead of atriangle or 4-sided polygon) the polynomial degree will be degree 19over degree 2, and for a 6-sided polygon, the polynomial degree will bedegree 23 over degree 2.

Accordingly, for an N-sided surface patch where, e.g., N>7 for thenumber of parametric polygonal sides for the patch, wherein the patchboundary curves are represented as in Formula (14) above, then such apatch can be effectively approximated as closely as desired by:

-   -   Decompose the polygonal domain of the patch into a plurality of        polygons, each having a smaller number of sides, e.g.,        preferably polygons having less than 5 sides; and    -   For each of the resulting polygons from step (a) immediately        above, generate a corresponding surface S(u,v) as described        above.        6.3 The G¹ Surface

Formula (12) above can be expanded in a straightforward way to includecontinuity of tangency at the boundaries between subpatches sharing acommon boundary curve. This is done by replacing the boundary curveƒ_(i) in Formula (12) with a lofted surface, or ribbon (RB) definedusing the boundary curve ƒ_(i) and a second curve g_(i), wherein thesurface between these two curves defines the tangency conditions at thecommon boundary curve between the subpatches. Such ribbons are describedin U.S. Pat. No. 7,236,167. In particular, for each of the subpatchessharing the common boundary curve, there is a corresponding ribbondefined by, and having a contour and extent, containing the tangents ofthe subpatch at the common boundary. Thus, when ribbons for differentsubpatches of a common boundary are continuous across this commonboundary, then these subpatches smoothly and continuous blend into oneanother.

Such a ribbon for a patch boundary curve c_(i) (e.g., represented byf_(i)(t_(i)) of Formula (14) above) may be defined as follows:RB(s _(i) ,t _(i))=(1−s _(i))θ_(i)(t _(i))+s _(i) g _(i)(t _(i))  (20)where t_(i) is the t-value parameter in Formula (18), and s_(i) isrelated linearly to the distance d_(i) in Formula (19), such that s_(i)varies between 0 and 1, e.g., s_(i) may be:

-   -   the same as distance d_(p)    -   its negative, or    -   in some cases, an affine transformation of d_(i).

If there is a ribbon for each boundary curve of a (sub)patch, the formof the (sub)patch satisfying desired tangency conditions of the ribbonsis:S(u,v)=Σ_(i=1,N)[RB(s _(i)(u,v)t _(i)(u,v))Π_(j=1,N,i≠j) W(d_(j)(u,v)  (21)

Note that Formula (21) adds one more degree in the numerator, giving:

-   -   Degree 16 over 2 for the 4-sided patch.    -   Degree 12 over 2 for the 3-sided patch.        6.4 Description of the Conversion of the Novel N-Sided Surface        Patches into Bezier and/or NURBS        The Algorithm:        Step 1: Expand numerator of (10) as a bivariate polynomial        (known art).        Step 2: Collect all the coefficients into a matrix M, so that U        M V is the polynomial of Step 1.        Step 3: Multiply M by PB on both sides—PB M PB, where PB is the        matrix of binomial coefficients.        Step 4: Do Steps 1-3 for the denominator.

The result of this process is a matrix of coefficients Bz that representthe Bezier coefficients of 10. From this conversion to B-splines isknown art.

The surface patch needs trimming by the surface functions which aremappings of the base polygon.

7. G2 Surface Models for N-Sided Patches

7.1 Patch Description

A requisite for a surface is a G¹ surface as one skilled in the art willunderstand. We therefore begin with a description of a G¹ surface, fromwhich we then describe techniques to create a G² surface.

We start by constructing a plurality of lofted surfaces, R_(i),according to the following formula:R _(i)(s _(i) ,t _(i))=(1−s _(i))ƒ_(i)(t _(i))+s _(i) g _(i)(t_(i)),  (22)

In Formula (22) hereinabove, the parameter s_(i) parameterizes thei^(th) loft, where 0≤s_(i)≤1, i.e., s_(i) is the distance to the i^(th)footprint f_(i)(t_(i)) with respect to u. Each g_(i)(t) is a user givenparametric curve that is used to define the boundary of the loft. Wecall the loft a “ribbon”. With s_(i) and t_(i) given as functions of u,we can give the following variation of Formula (22) to obtain thefollowing formula used for computing a G¹ surface:F(u)=Σ_(i)[R _(i)(s _(i) ,t _(i)))](W _(i)(u)/Σ_(i) W _(i)(u))²  (23)wherein W_(i)(u) is (1/s_(i))^(p), p an exponent, and Formula (23)contains Formula (22)

We modify Formula (23) for computing a G² surface by replacing the curveƒ_(i) with a new “ribbon” function Q_(i)(t_(i)(u,v)) resulting inS(u,v)=Σ_(i=1,N)[Q _(i)(t _(i)(u,v))Π_(j=1,N) ,W(d _(j)(u,v)/Σ_(i=1,N)W(d _(j)(u,v))  (24)where the functions d_(j)(u,v) are distance-like functions in parameterspace from the j^(th) parameter space polygonal edge, the functionst_(j)(u,v) are parameter functions, both described in U.S. Pat. No.7,236,167, and wherein a possible additional weight function

(x) may be:

(x)=x ^(n),  (25)Where n>2, and a function for each of the boundary curves ƒ_(i) throughwhich the surface S(u,v) must pass is:Q _(i)(t)=(1−t)²ƒ_(i)+(1−t)tg _(i) +t ² h _(i)  (26)with points {p_(i,0), p_(i,1), p_(i,2), p_(i,3)} for defining the i^(th)boundary curve ƒ_(i) of a patch (in object space wherein the i^(th)boundary curve and these points are the end points of, e.g., profilehandles as disclosed in U.S. Pat. No. 7,236,167.Two further user given curves g_(i) and h_(i) are constructed in afashion similar to ƒ_(i) in such a way as to give a ribbon surfaceQ_(i)(t_(i)(u,v)) with a parabolic cross-section

The “t-value” function of (1) may be defined as followst _(i)(u,v)=((s _(x) −u)(s _(x) −e _(x))+(s _(y) −e _(y)))/((e _(x) −s_(x))²+(e _(y) −s _(y))²)  (27)

Note that Formula (25) computes t_(i)(u,v) to be the parameterizationvalue (between 0 and 1) for a point p(u,v) on the i^(th) side of thebase polygon in parameter space, wherein p(u,v) is the perpendiculardrop point or projection of (u,v) onto the side of the base polygon.Moreover, t_(i) is linear in u and v.

Various alternative computations for t_(i)(u,v) are also within thescope of the present disclosure. In particular, t_(i)(u,v) may bealternatively defined as follows:t _(i)(u,v)=|(mu+b)−v|, or, t _(i)(u,v)=|(v−b)/m−u|.

The “distance” function d_(i)(u,v) may be defined as follows:d _(i)(u,v)=(mv+b−u)sin(arctan(m)).  (28)Various alternative computations for d_(i)(u,v) are within the scope ofthe present disclosure. In particular, d_(i)(u,v)=[(mv+b−u)sin(arctan(m))]² for m not vertical, and (b−u)² otherwise.7.2 Results

If two patches abut and their ribbon curves have G² continuity as shownin FIG. 54, then it can be shown that the surfaces will also have G²continuity.

FIGS. 55A and 55B compare the effect of the G¹ embodiment (the left autobody) to the G² embodiment (the right auto body). One will note thesmoother appearance of the model with G² computation.

FIGS. 56A and 56B show yet another comparison of G¹ and G², wherein theleft auto body front (FIG. 56A) was computed with G¹ and the right autobody front (FIG. 56B) was computed with G².

7.3 Defining the Ribbons

Lastly we discuss methods of defining the ribbons so that there are G²continuous as needed to make the surfaces G² continuous.

The particular method to define lofts in general with the requisitetangents or curvatures depends on the curve types chosen as boundarycurves. we used B-spline curve boundaries where the control points ofthe lofts are easily defined by offsetting the attribute curves controlpoints b_(i) with symmetric displacement vectors as in the individualcase in FIG. 57A. If the rt_(i) and lt_(i) are anti-parallel itguarantees G¹ continuity. If the parabolas defined by crt_(i), rt_(i)and b_(i) have the same curvature as clt_(i), lt_(i) and b_(i) for alli, then the patches will meet with G² continuity. Similar techniquesapply to other parametric curves like Bezier, etc. FIG. 57B shows thistechnique applied to all the control points along a curve. The foregoingdiscussion of the invention has been presented for purposes ofillustration and description. Further, the description is not intendedto limit the invention to the form disclosed herein. Consequently,variation and modification commiserate with the above teachings, withinthe skill and knowledge of the relevant art, are within the scope of thepresent invention. The embodiment described hereinabove is furtherintended to explain the best mode presently known of practicing theinvention and to enable others skilled in the art to utilize theinvention as such, or in other embodiments, and with the variousmodifications required 55 by their particular application or uses of theinvention.

8. Applications

A computational system according to the present disclosure may be usedin a large number of computational design domains. In particular, thefollowing list provides brief descriptions of some of the areas wheresuch a computational system can be applied.

8.1 Container Design

Free-form design of containers such as bottles has been heretoforenon-intuitive and tedious. The present disclosure shows how to alleviatethese drawbacks.

8.2 Automotive Design

In the automotive industry, the present disclosure can be used for autobody design as well as for auto component design. In particular, theease with which deformations of parts and contours can be performed witha computational system according to the present disclosure allows forstraightforward deformation of components and recesses so that thefitting of components in particular recesses is more easily designed.

8.3 Aerospace

The present disclosure provides high precision trimming and surfacepatching operations which are required by the aerospace industry.

8.4 Shipbuilding

Unique to the shipbuilding industry is the need for the design of shiphulls and propellers. Designs of both hulls and propellers may be drivenby the physics of the constraints related to water flow. Thesatisfaction of such constraints can be incorporated into acomputational system according to the present disclosure.

8.5 Traditional CAD/CAM Applications

Applications for the design of engines, piping layouts and sheet metalproducts typically require trimming and blending capabilities. Thus,since the present disclosure is particularly efficient at providing suchoperations as well as providing easy deformations of surfaces, itseffectiveness in these areas may be of particular merit.

8.6 Other Applications

The following is a list of other areas where the present disclosure maybe used for computational design. These are: home electronic andappliance design, plastic injection mold design, tool and die design,toy design, geological modeling, geographical modeling, mining design,art and entertainment, animation, sculpture, fluid dynamics,meteorology, heat flow, electromagnetics, plastic surgery, burn masks,orthodontics, prosthetics, clothing design, shoe design, architecturaldesign, virtual reality design, scientific visualization of data,geometric models for training personnel (e.g., medical training).

The foregoing discussion of the invention has been presented forpurposes of illustration and description. Further, the description isnot intended to limit the invention to the form disclosed herein.Consequently, variations and modifications commensurate with the aboveteachings, within the skill and knowledge of the relevant art, arewithin the scope of the present disclosure. The embodiment describedhereinabove is further intended to explain the best mode presently knownof practicing the invention and to enable others skilled in the art toutilize the invention as such, or in other embodiments, and with thevarious modifications required by their particular application or usesof the invention. It is intended that the appended claims be construedto include alternative embodiments to the extent permitted by the priorart.

Appendix A—Matlab Code for Patches

The following encodes the previous mathematics with MATLAB, and was usedto generate the images.

% Lowest possible degree(?) power surface for triangular and rectangularpatches. function gone = Convert( ) cla reset; axis vis3d;light(‘Position’,[−50 0 50],‘Style’,‘infinite’); hold on step = 0.05; %For display resolution % Triangular patch for v = 0:step:1−step for u =0:step:(1−v−step/2) % 4 corners of the display rectangle, out of whichwe make 2 triangles bl = PtoF(1,u,v); br = PtoF(1,u+step,v); % Findvalue on surface ul = PtoF(1,u,v+step); ur = PtoF(1,u+step,v+step); M1 =[bl;ul;br]; M2 = [ul;ur;br]; %Make display trianglesfill3(M1(:,1),M1(:,2),M1(:,3),‘g’,‘EdgeColor’,‘none’); % shadefill3(M2(:,1),M2(:,2),M2(:,3),‘r’,‘EdgeColor’,‘none’); end end %Rectangular patch for v = 0:step:1−step for u = 0:step:1−step % 4corners of a display rectangle, out of which we make 2 triangles bl =PtoF(2,u,v); br = PtoF(2,u+step,v); % Find value on surface ul =PtoF(2,u,v+step); ur = PtoF(2,u+step,v+step); M1 = [bl;ul;br]; M2 =[ul;ur;br]; % Make display trianglesfill3(M1(:,1),M1(:,2),M1(:,3),‘g’,‘EdgeColor’,‘none’); % shadefill3(M2(:,1),M2(:,2),M2(:,3),‘r’,‘EdgeColor’,‘none’); end end % end offunction Convert function vert = PtoF(pat_idx,u,v) %Computes Point onFunction nm_sum = 0; dnm_sum = 0; switch pat_idx case 1 ft_num = 3;%Triangle case 2 ft_num = 4; %Rectangle end for i = 1:ft_num % Finalweighted sum nm_sum =nm_sum+Weight(pat_idx,i,u,v)*Ribbon(pat_idx,i,u,v); dnm_sum =dnm_sum+Weight(pat_idx,i,u,v); end vert = nm_sum/dnm_sum; % end offunction PtoF % Cubic Rock Function − pi are control points function pt= RockCub(p1,p0,p3,p2,t) %different order in arguments than Bezier Lt =(1−t)*p1+t*p0; Rt=(1−t)*p3+t*p2; %left and right tangents pt =(1−RW(t))*Rt + RW(t)* Lt; %end of RockCub function % Distance functionfor base polygons − triangle and rectangle function dt =DisTri(ft_idx,u,v) switch ft_idx case 1 dt = v; case 2 dt = u; case 3 %given a line v=mu+b, the distance from (u,v) to the line is % (mv + b −u)*sin(arctan(m)) We choose m = −1, b = 1 dt = (−v + 1 − u)*0.707; end %of function DisTri function t = Trit(ft_idx,u,v) %t value on triangleswitch ft_idx case 1 t = 1−u; case 2 t = v; case 3 t = (1+u−v)/2; %Expects line as in DisTri end % of function Trit function t =Rect(ft_idx,u,v) %t value on rectangle switch ft_idx case 1 t = 1−u;case 2 t = v; case 3 t = u; case 4 t = 1−v; end % of function Rectfunction dt = DisRec(ft_idx,u,v) switch ft_idx case 1 dt = v; case 2 dt= u; case 3 dt = 1−v; case 4 dt = 1−u; end % of function DisRec functionwt = Weight(pat_idx,ft_idx,u,v) wt = 1.0; switch pat_idx %Weights as inFreeDimension case 1 for i=1:3 d = DisTri(i,u,v); if i == ft_idx dfac =1−d{circumflex over ( )}2; else dfac = d{circumflex over ( )}2; end %ifwt = wt*dfac; end %for case 2 for i=1:4 d = DisRec(i,u,v); if i ==ft_idx dfac = 1−d{circumflex over ( )}2; else dfac = d{circumflex over( )}2; end %if wt = wt*dfac; end %for end %switch % end of functionWeight function wt = RW(t) %Rational weight function wt =(t−1){circumflex over ( )}2/(2*t{circumflex over ( )}2 − 2*t + 1); %endof function RW % Define curves and ribbon boundaries function rib =Ribbon(pat_idx,ft_idx,u,v) switch pat_idx case 1 %triangle A = [100 0−90; 0 0 0; −70.7 70.7 −90]; % Curves B = [100 0 0; −35 35 0; 20 150−90] C = [50 0 0; −70.7 70.7 0; 100 120 −90]; D = [0 0 0; −70.7 70.7−90; 100 0 −90]; RA = [0 100 0; 100 100 0; 0 0 60]; % Boundaries RB = [0100 0; 100 100 0; 0 0 60]; RC = [0 100 0; 100 100 0; 0 0 60]; RD = [0100 0; 100 100 0; 0 0 60]; dt = DisTri(ft_idx,u,v); t =Trit(ft_idx,u,v); case 2 %rectangle A = [0 −100 −90; 0 0 0; 100 0−90;100 −100 −90]; % Curves B = [0 −100 0; 50 0 0; 100 −33 −90;67 −100−90]; C = [0 −50 0; 100 0 0; 100 −67 −90;33 −100 −90]; D = [0 0 0; 100 0−90; 100 −100 −90;0 −100 −90]; RA = [50 0 0; 0 −90 0; 0 0 60;0 0 60]; %Boundaries RB = [50 0 0; 0 −90 0; 0 0 60;0 0 60]; RC = [50 0 0; 0 −90 0;0 0 60;0 0 60]; RD = [50 0 0; 0 −90 0; 0 0 60;0 0 60]; dt =DisRec(ft_idx,u,v); t = Rect(ft_idx,u,v); end %switch AR = A+RA; BR =B+RB; CR = C+RC; DR = D+RD; %Ribbons computed by offsetting curve a =A(ft_idx,:); b = B(ft_idx,:); %Curve c = C(ft_idx,:); d = D(ft_idx,:);ar = AR(ft_idx,:); br = BR(ft_idx,:); % Ribbon boundary cr =CR(ft_idx,:); dr = DR(ft_idx,:); rib =(1−dt)*RockCub(a,b,c,d,t)+dt*RockCub(ar,br,cr,dr,t); %Blend of curve andribbon % end of function RibbonConversion of FD Surface to BezlerThe following is an example of a 4-sided FreeDimension (FD) surfacewithout tangency. to a Bezier surface.The four sided FreeDimension (FD) surface was defined with 16 3D inputs,four on each curve. They are, for the ith curve:msi=marker startmei=marker endchsi=curve handle startchei=curve handle endThe function has a numerator and denominator that are separatelyconverted to Bezier form. The denominator is degree 4 in u and v. Justto get these things archived, we first write the FD denominator inmatrix form, i.e. polynomial bases times the matrix gives us thefunction. The matrix is:DN:=Matrix(5,5,{(1,1)=−4,(1,2)=8,(1,3)=−2,(1,4)=−2,(1,5)=1,(2,1)=8,(2,2)=−16,(2,3)=4,(2,4)=4,(2,5)=−2,(3,1)=−2,(3,2)=4,(3,3)=0,(3,4)=−2,(3,5)=1,(4,1)=−2,(4,2)=4,(4,3)=−2,(4,4)=0,(4,5)=0,(5,1)=1,(5,2)=−2,(5,3)=1,(5,4)=0,(5,5)=0})  (1)The conversion matrix that takes DN to Bezier form isMatrix(5,5),{(1,1)=0,(1,2)=0,(1,3)=⅙,(1,4)=0,(1,5)=0,(2,1)=0,(2,2)=0,(2,3)=1/12,(2,4)=0,(2,5)=0,(3,1)=⅙,(3,2)= 1/12,(3,3)=0,(3,4)=1/12,(3,5)=⅙,(4,1)=0,(4,2)=0,(4,3)=1/12,(4,4)=0,(4,5)=0,(5,1)=0,(5,2)=0,(5,3)=⅙,(5,4)=0,(5,5)=0))  (2)That is, one multiplies DN left and right by t (2) to get a new matrix,which when multiplied by the Bernstein polynomials gives a 4 by 4 Beziersurface that evaluates to the same denominator as the original form. ItisFDB4:=(1−u){circumflex over ( )}4*(1−){circumflex over( )}2*v{circumflex over ( )}2+2*(1−u){circumflex over( )}3*u*(1−v){circumflex over ( )}2*v{circumflex over( )}2+6*(1−u){circumflex over ( )}2*u{circumflex over( )}2*(⅙*(1−v){circumflex over ( )}4+⅓*(1−v){circumflex over( )}3*v+(⅓)*(1−v)*v{circumflex over ( )}3+⅙*v{circumflex over( )}4)+2*(1−u)*u{circumflex over ( )}3*(1−v){circumflex over( )}2*v{circumflex over ( )}2+{circumflex over ( )}4*(1−v){circumflexover ( )}2*v{circumflex over ( )}  (3)We now do the same things to the numerator. The matrix for the numeratorin polynomial form is[[0,0, 1/21me1, 3/35che1, 3/35chs1, 1/21ms1,0,0],[0,0, 5/147me1,3/49che1, 3/49chs1, 5/147ms1,0,0],[ 1/21ms2, 5/147ms2, 1/49me1+1/49ms2,1/105ms2+ 2/735me4+ 9/245che1, 2/735ms2+ 1/105me4+ 9/245chs1,1/49ms1+ 1/49me4,5/147me4,1/21me4],[ 3/35chs2, 3/49chs2, 1/105me1+2/735ms3+ 9/245chs2, 3/175che1+ 3/175chs2+ 6/1225chs3+ 6/1225che4,3/175chs1+ 6/1225chs2+ 6/1225che3+ 3/175che4, 9/245che4+ 1/105ms1+2/735me3, 3/49che4, 3/35che4],[ 3/35che2, 3/49che2, 2/735me1+ 1/105ms3+9/245che2, 3/175chs3+ 6/1225chs4+ 6/1225che1+ 3/175che2, 3/175che3+3/175chs4+ 6/1225chs1+ 6/1225che2, 9/245chs4+ 2/735ms1+ 1/105me3,3/49chs4, 3/35chs4],[ 1/21me2, 5/147me2, 1/49ms3+ 1/49me2, 9/245chs3+1/105me2+ 2/735ms4, 9/245che3+ 2/735me2+ 1/105ms4, 1/49me3+ 1/49ms4,5/147ms4, 1/21ms4],[0,0, 5/147ms3, 3/49chs3, 3/49che3,5/147me3,0,0],[0,0, 1/21ms3, 3/35chs3, 3/35che3, 1/21me3,0,0]]  (4)Notice that (4) has the vector control points msi, etc. The next matrixis converted to Bezier form. In other words, they are Bezier controlpoints of the numerator, which could be used in an IGES file fortransfer.[[0,0, 1/21me1, 3/35che1, 3/35chs1, 1/21ms1,0,0],[0,0, 5/147me1,3/49che1, 3/49chs1, 5/147ms1,0,0],[ 1/21ms2, 5/147ms2, 1/49me1+ 1/49ms2,1/105ms2+ 2/735me4+ 9/245che1, 2/735ms2+ 1/105me4+ 9/245chs1, 1/49ms1+1/49me4, 5/147me4, 1/21me4],[ 3/35chs2, 3/49chs2, 1/105me1+ 2/735ms3+9/245chs2, 3/175che1+ 3/175chs2+ 6/1225chs3+ 6/1225che4, 3/175chs1+6/1225chs2+ 6/1225che3+ 3/175che4, 9/245che4+ 1/105ms1+ 2/735me3,3/49che4, 3/35che4],[ 3/35che2, 3/49che2, 2/735me1+ 1/105ms3+ 9/245che2,3/175chs3+ 6/1225chs4+ 6/1225che1+ 3/175che2, 3/175che3+ 3/175chs4+6/1225chs1+ 6/1225che2, 9/245chs4+ 2/735ms1+ 1/105me3, 3/49chs4,3/35chs4],[ 1/21me2, 5/147me2, 1/49ms3+ 1/49me2, 9/245chs3+ 1/105me2+2/735ms4, 9/245che3+ 2/735me2+ 1/105ms4, 1/49me3+ 1/49ms4,5/147ms4,1/21ms4],[0,0, 5/147ms3, 3/49chs3, 3/49che3, 5/147me3,0,0],[0,0,1/21ms3, 3/35chs3, 3/35che3, 1/21me3,0,0]].  (5)

The final function is the Bezier surface obtained by multiplying (5) bythe Bernstein polynomials.FDB7:=(1−u){circumflex over ( )}7*((1−v){circumflex over( )}5*v{circumflex over ( )}*ms2+3*(1−v){circumflex over( )}4*v{circumflex over ( )}3*chs2+3*(1−v){circumflex over( )}3*v{circumflex over ( )}4*che2+(1−v){circumflex over( )}2*v{circumflex over ( )}5*me2)+7*(1−u){circumflex over ( )}6*u*(5/7*(1−v){circumflex over ( )}5*v{circumflex over( )}2*ms2+15/7*(1−v){circumflex over ( )}4*v{circumflex over( )}3*cbs2+15/7*(1−v){circumflex over ( )}3*v{circumflex over( )}4*che2+ 5/7*(1−v){circumflex over ( )}2*v{circumflex over( )}5*me2)+21*(1−u){circumflex over ( )}5*u{circumflex over ( )}2*(1/21*(1−v){circumflex over ( )}7*me1+ 5/21*(1−v){circumflex over( )}6*v*me1+21*(1−v){circumflex over ( )}v{circumflex over ( )}2*(1/49*me1+ 1/49*ms2)+35*(1−v){circumflex over ( )}4*v{circumflex over( )}3*( 1/105*me1+ 2/735*ms3+ 9/245*cbs2)+35*(1−v){circumflex over( )}3*v{circumflex over ( )}4*( 2/735*me1+ 1/105*ms3+9/245*che2)+21*(1−v){circumflex over ( )}2*v{circumflex over ( )}5*(1/49*ms3+ 1/49*me2)+( 5/21)*(1−v)*v{circumflex over ( )}6*ms3+1/21*v{circumflex over ( )}7*ms3)+35*(1−u){circumflex over( )}4*u{circumflex over ( )}3*( 3/35*(1−v){circumflex over ( )}7*che1+3/7*(1−v){circumflex over ( )}6*v*che1+21*(1−v){circumflex over( )}5*v{circumflex over ( )}2*( 1/105*ms2+ 2/735*me4+9/245*che1)+35*(1−v){circumflex over ( )}4*v{circumflex over ( )}3*(3/175*che1+ 3/175*chs2+ 6/1225*chs3+ 6/1225*che4)+35*(1−v){circumflexover ( )}3*v{circumflex over ( )}4*( 3/175*cs3+ 6/1225*chs4+ 6/1225*me1+3/175*che2)+21*(1−v){circumflex over ( )}2*v{circumflex over ( )}5*(9/245*ch3+ 1/105*me2+ 2/735*ms4)+( 3/7)*(1−v)*v{circumflex over( )}6*cbs3+ 3/35*v{circumflex over ( )}7*chs3)+35*(1−u){circumflex over( )}3*u{circumflex over ( )}4*( 3/35*(1−v){circumflex over ( )}7*chs1+3/7*(1−v){circumflex over ( )}6*v{circumflex over( )}chs1+21*(1−v){circumflex over ( )}5*v{circumflex over ( )}2*(2/735*ms2+ 1/105*me4+ 9/245*chs1)+35*(1−v){circumflex over( )}4*v{circumflex over ( )}3*( 3/175*chs1+ 6/1225*chs2+ 6/1225*che3+3/175*che4)+35*(1−v){circumflex over ( )}3*v{circumflex over ( )}4*(3/175*che3+ 3/175*chs4+ 6/1225*chs1+ 6/1225*che2)+21*(1−v){circumflexover ( )}2*v{circumflex over ( )}5*( 9/245*che3+ 2/735*me2+ 1/105*ms4)+(3/7)*(1−v)*v{circumflex over ( )}6*che3+ 3/35*v{circumflex over( )}7*che3)+21*(1−u){circumflex over ( )}2*u{circumflex over ( )}5*(1/21*(1−v){circumflex over ( )}7*ms1+ 5/21*(1−v){circumflex over( )}6*v*ms1+21*(1−v){circumflex over ( )}5*v{circumflex over ( )}2*(1/49*ms1+ 1/49*me4)+35*(1−v){circumflex over ( )}4*v{circumflex over( )}3*( 9/245*che4+ 1/105*ms1+ 2/735*me3)+35*(1−v){circumflex over( )}3*v{circumflex over ( )}4*( 9/245*chs4+ 2/735*ms1+1/105*me3)+21*(1−v){circumflex over ( )}2*v{circumflex over ( )}5*(1/49*me3+ 1/49*ms4)+( 5/21)*(1−v)*v{circumflex over ( )}6*me3+1/21*v{circumflex over ( )}7*me3)+7*(1−u)*u{circumflex over ( )}6*(5/7*(1−v){circumflex over ( )}5*v{circumflex over( )}2*me4+15/7*(1−v){circumflex over ( )}4*v{circumflex over( )}3*che4+15/7*(1−v){circumflex over ( )}3*v{circumflex over( )}4*chs4+ 5/7*(1−v){circumflex over ( )}2*v{circumflex over( )}5*ms4)+u{circumflex over ( )}7*((1−v){circumflex over( )}5*v{circumflex over ( )}2*me4+3*(1−v){circumflex over( )}4*v{circumflex over ( )}3*che4+3*(1−v){circumflex over( )}3*v{circumflex over ( )}4*chs4+(1−v){circumflex over( )}2*v{circumflex over ( )}5*m4)  (6)It is easy to see from (6) that the degree of the numerator is 7 in bothvariables. The degree of the denominator for the Bezier form in (3) isonly 4. Conversion of Bezier to B-spline is reasonably straightforwardand changes only the knot vector. The control points are the same.Adding the tangent form doubles the number of control points, but onlyadds one degree to the numerator; thus degree 8 over degree 4. Ingeneral, adding a side to the surface will add 2 degrees top and bottomto the function.

What is claimed is:
 1. A computer-implemented method for processing adeformation of a three dimensional surface in substantially real time,the deformation resulting from processing a series of selectioninstructions for the deformation, comprising performing the following bycomputer equipment: displaying, on a computer-controlled display, avisual representation of at least one geometric surface represented inthe computer equipment using at least three parameters, each parameterfor representing the at least one geometric surface in a distinct one ofthree independent spatial coordinates, the at least one geometricsurface represented, in the computer equipment, by first data providinga parameterization of the at least one geometric surface (P_(S)), theparameterization P_(S) having an at least two dimensional extent as apre-image; accessing respective data representing one or more additionalgeometric objects used in determining shape modified instances of the atleast one geometric surface, each of the one or more additionalgeometric objects: (i) has a corresponding portion that is coincidentwith the at least one geometric surface, and (ii) has a parameterization(P_(O)), that is twice differentiable at the corresponding portion,wherein a first derivative of the parameterization P_(S) at thecorresponding portion agrees with a first derivative of theparameterization P_(O) at the corresponding portion, wherein a secondderivative of the parameterization P_(S) at the corresponding portionagrees with a second derivative of the parameterization P_(O) at thecorresponding portion, and wherein a third derivative of theparameterization P_(S) at the corresponding portion does not agree witha third derivative of the parameterization P_(O) at the correspondingportion; for at least one point of a plurality of points, each of thepoints included in the corresponding portion of one of the additionalgeometric objects, selecting: (a) the at least one point, or (b)associated information providing geometric information characteristic ofa corresponding one of the additional geometric objects having the atleast one point; iteratively performing A1 through A4 following forproviding a user perceived substantially real time deformation of the atleast one geometric surface during a processing of the series ofselection instructions by the computer equipment, wherein each of theselection instructions corresponds to a change to one of: a location forthe at least one point, or the associated information; (A1) receiving anext one of the selection instructions by the computer equipment; (A2)deriving, using the next one of the selection instructions, datarepresenting a modified version of the corresponding one additionalgeometric object, wherein for another point, for the corresponding oneadditional geometric object, the another point being also one of theplurality of points such that the another point is not selected forcontributing to the user perceived real time deformation, and theanother point has second associated information providing a secondgeometric information characteristic of the corresponding one additionalgeometric object at the another point, then at least one of: (1) theanother point, and (2) the second geometric information characteristic,remains unchanged and also corresponds with the modified version; (A3)subsequently, determining surface data representing one of the shapemodified instances of the at least one geometric surface, the surfacedata determined using data representing the modified version of thecorresponding one additional geometric object; and (A4) displaying theone shape modified instance of the at least one geometric surface on thecomputer-controlled display.
 2. The method of claim 1, wherein thegeometric information characteristic includes a corresponding relatedextent for the at least one point, wherein one or more of: i) thecorresponding related extent has a parametric direction substantiallyidentical to a parametric tangency direction of the at least onegeometric surface at the at least one point according to theparameterization of the at least one geometric surface, and ii) a lengthof the corresponding related extent for the at least one point isrepresentative of how closely a contour of the at least one geometricsurface follows the parametric tangency direction as the contour extendsaway from the at least one point.
 3. The method of claim 1, furtherincluding: adding an additional boundary to the at least one geometricsurface, or to a specified one of the shape modified instances, theadditional boundary being responsive to a corresponding selectioninstruction; subsequently, generating a further geometric object for theadditional boundary; and subsequently, iteratively performing the A1through A4 using the further geometric object as one of the additionalgeometric objects.
 4. The method of claim 3, wherein the adding theadditional boundary includes: processing the corresponding selectioninstruction by the computer equipment for identifying a point, q, forthe at least one geometric surface or the specified one shape modifiedinstance; and subsequently, generating the additional boundary, whereina boundary of the at least one geometric surface is modified toconnectedly extend between the point q and another point for the atleast one geometric surface such that the additional boundary isinterpolated from the at least one geometric surface or the specifiedone shape modified instance so that a tolerance between the at least onegeometric surface or the specified one shape modified instance and theadditional boundary is below a predetermined maximum tolerance.
 5. Themethod of claim 1, wherein the determining the surface data representingthe one shape modified instance includes combining a plurality of termsfor determining points of the one shape modified instance, wherein foreach pre-image point in a plurality of points distributed throughout theat least two dimensional extent, and for the modified version,conditions (I) and (II) following hold: (I) there are one or morecorresponding terms, T, of the plurality of the terms, each term, T,dependent on the pre-image point and the modified version; and (II) eachof the corresponding terms T is determined by computing a multiplicativeproduct of: (a) a corresponding term weighting, and (b) datarepresenting a particular point of the modified version, wherein theparticular point has the pre-image point, upon which the correspondingterm T is dependent, as a pre-image; wherein for each term, T′, of aplurality of the one or more corresponding terms, the corresponding termweighting (W) for the term T′, is determined by the computer equipmentcomputing a value determined using a distance-like measurement between:(i) the pre-image point that the term T′ is dependent upon, and (ii) apre-image of the corresponding portion and the corresponding oneadditional geometric object.
 6. The method of claim 5, wherein each ofthe one or more additional geometric objects has a pre-image, andfurther including, for each term, the corresponding term weighting (Wtg)used in determining the terms T by the computer equipment, obtaining acorresponding value of a blending function, wherein the correspondingvalue is used for determining the corresponding term weighting Wtg. 7.The method of claim 5, wherein, for at least one of the correspondingterms T′, the corresponding term weighting, W, vary monotonically with acorresponding distance-like measurements, each of the correspondingdistance-like measurements, DM, being indicative of a spacing between:(i) the pre-image point upon which at least one corresponding of theterms T′ depends, and (ii) a pre-image of the corresponding portion andof the corresponding one additional geometric object.
 8. The method ofclaim 5, wherein the distance-like measurement is determined by adistance-like function.
 9. The method of claim 1, wherein the next oneof the selection instructions moves the at least one point along apredetermined boundary curve, C, containing the at least one point, andfurther including changing the one shape modified instance of the atleast one geometric surface as a result of the next one of the selectioninstructions.
 10. The method of claim 1, wherein the geometricinformation characteristic includes a corresponding related extent, E,for the corresponding one additional geometric object, and furtherincluding maintaining a predetermined constraint for the correspondingrelated extent E, wherein the constraint includes one of: (a)constraining the extent E to a predetermined range of directions; (b)constraining the extent E to a predetermined range of magnitudes; and(c) constraining the extent E to a predetermined range of curvatures.11. The method of claim 1, wherein the geometric informationcharacteristic includes data for a corresponding related extent, E, forthe corresponding one additional geometric object, and further includingmaintaining a predetermined constraint for the corresponding relatedextent E, wherein the constraint includes one of: (a) constraining theextent E to lie in a plane with another related extent for some otherpoint of the corresponding portion for at least one of the additionalgeometric objects; and (b) constraining the extent E to transformidentically with another related extent for some other point of thecorresponding portion for at least one of the additional geometricobjects.
 12. The method of claim 1, wherein the corresponding portion isa connected boundary of the at least one geometric surface, theconnected boundary including a plurality of boundary points.
 13. Themethod of claim 1, wherein the corresponding one additional geometricobject is a ruled surface.
 14. The method of claim 1, wherein thegeometric information characteristic includes data indicative of anextent of at least a portion of the corresponding one additionalgeometric object.
 15. The method of claim 1, wherein the geometricinformation characteristic includes data indicative of a shape of atleast a portion of the corresponding one additional geometric object.16. The method of claim 1, wherein the corresponding portion includes aconnected portion that connects the at least one point to the anotherpoint.
 17. The method of claim 1, wherein the corresponding portionincludes a curve between the at least one point to the another point.18. The method of claim 1, wherein for each of one or more of theselection instructions, step of receiving input from a user pointing orselection device.
 19. The method of claim 1, wherein for each of one ormore of the selection instructions (SS), step of computing, by thecomputer equipment, a difference between the selection instruction, SS,and a corresponding instance of the next one of the continuous selectioninstructions.
 20. The method of claim 1, wherein the processing thedeformation of the three dimensional surface comprises animating thethree dimensional surface.
 21. A non-transitory computer-readable mediumcomprising computer executable code for performing the method ofclaim
 1. 22. The method of claim 1, wherein each of the one or moreadditional geometric objects is represented by a surface.
 23. The methodof claim 1, wherein, for each of the one or more additional geometricobjects, the corresponding portion includes a join with the at least onegeometric surface.
 24. A computer-implemented method for processing adeformation of a three dimensional surface in substantially real time,the deformation resulting from processing a series of selectioninstructions for the deformation, comprising performing the following bycomputer equipment: displaying, on a computer-controlled display, avisual representation of at least one geometric surface represented inthe computer equipment using at least three parameters, each parameterfor representing the at least one geometric surface in a distinct one ofthree independent spatial coordinates, the at least one geometricsurface having a parameterization of an at least two dimensional extentas a pre-image; accessing respective data representing one or moreadditional geometric objects used in determining shape modifiedinstances of the at least one geometric surface, each of the one or moreadditional geometric objects: (i) has a corresponding portion that iscoincident with the at least one geometric surface, and (ii) has aparameterization that is twice differentiable at the correspondingportion, wherein each of the one or more additional geometric objects isG² geometric continuous at the corresponding portion with the at leastone geometric surface, and wherein at least one of the one or moreadditional geometric objects is not G³ geometric continuous with the atleast one geometric surface at the corresponding portion; for at leastone point of a plurality of points, each of the points included in thecorresponding portion of one of the additional geometric objects,selecting: (a) the at least one point, or (b) associated informationproviding geometric information characteristic of a corresponding one ofthe additional geometric objects having the at least one point;iteratively performing A1 through A4 following for providing a userperceived substantially real time deformation of the at least onegeometric surface during a processing of the series of selectioninstructions by the computer equipment, wherein each of the selectioninstructions corresponds to a change to one of: a location for the atleast one point, or the associated information; (A1) receiving a nextone of the selection instructions by the computer equipment; (A2)deriving, using the next one of the selection instructions, datarepresenting a modified version of the corresponding one additionalgeometric object, wherein for another point, for the corresponding oneadditional geometric object, the another point being also one of theplurality of points such that the another point is not selected forcontributing to the user perceived real time deformation, and theanother point has second associated information providing a secondgeometric information characteristic of the corresponding one additionalgeometric object at the another point, then at least one of: (1) theanother point, and (2) the second geometric information characteristic,remains unchanged and also corresponds with the modified version; (A3)subsequently, determining surface data representing one of the shapemodified instances of the at least one geometric surface, the surfacedata determined using data representing the modified version of thecorresponding one additional geometric object; and (A4) displaying theone shape modified instance of the at least one geometric surface on thecomputer-controlled display.
 25. A non-transitory computer-readablemedium comprising computer executable code for performing the method ofclaim
 24. 26. A computer-implemented method for processing a deformationof a three dimensional surface in substantially real time, thedeformation resulting from processing a series of selection instructionsfor the deformation, comprising performing the following by computerequipment: displaying, on a computer-controlled display, a visualrepresentation of at least one geometric surface represented in thecomputer equipment using at least three parameters, each parameter forrepresenting the at least one geometric surface surface in a distinctone of three independent spatial coordinates, the at least one geometricsurface represented, in the computer equipment, by first data providinga parameterization of the at least one geometric surface, theparameterization having an at least two dimensional extent as apre-image; accessing respective data representing one or more additionalgeometric objects used in determining shape modified instances of the atleast one geometric surface, each of the one or more additionalgeometric objects having a corresponding portion that is coincident withthe at least one geometric surface, wherein each of the one or moreadditional geometric objects is G^(k) geometric continuous with the atleast one geometric surface at the corresponding portion, and wherein atleast one of the one or more additional geometric objects is not G^(k+1)geometric continuous with the at least one geometric surface at thecorresponding portion, wherein k is dependent upon one or more criteriafor optical smoothness of at least one of the shape modified instances;for at least one point of a plurality of points, each of the pointsincluded in the corresponding portion of one of the additional geometricobjects, selecting: (a) the at least one point, or (b) associatedinformation providing geometric information characteristic of acorresponding one of the additional geometric objects having the atleast one point; iteratively performing A1 through A4 following forproviding a user perceived substantially real time deformation of the atleast one geometric surface during a processing of the series ofselection instructions by the computer equipment, wherein each of theselection instructions corresponds to a change to one of: a location forthe at least one point, or the associated information; (A1) receiving anext one of the selection instructions by the computer equipment; (A2)deriving, using the next one of the selection instructions, datarepresenting a modified version of the corresponding one additionalgeometric object, wherein for another point, for the corresponding oneadditional geometric object, the another point being also one of theplurality of points such that the another point is not selected forcontributing to the user perceived real time deformation, and theanother point has second associated information providing a secondgeometric information characteristic of the corresponding one additionalgeometric object at the another point, then at least one of: (1) theanother point, and (2) the second geometric information characteristic,remains unchanged and also corresponds with the modified version; (A3)subsequently, determining surface data representing one of the shapemodified instances of the at least one geometric surface, the surfacedata determined using data representing the modified version of thecorresponding one additional geometric object; and (A4) displaying theone shape modified instance of the at least one geometric surface on thecomputer-controlled display.
 27. A non-transitory computer-readablemedium comprising computer executable code for performing the method ofclaim
 26. 28. The method of claim 26, wherein the method is used for oneor more of: container design, vehicle design, aerospace design, shipdesign, CAD/CAM application, electronics device design, appliancedesign, plastic injection mold design, tool and die design, toy design,geological modeling, geographical modeling, mining design, gaming model,animation, sculpture, fluid dynamics design, heat flow, plastic surgery,orthodontics design, prosthetics design, clothing design, shoe design,architectural design, virtual reality design, scientific visualizationof data, and geometric models for training personnel.
 29. The method ofclaim 1, wherein each of the one or more additional geometric objects iscontinuous.
 30. The method of claim 1, wherein each of the one or moreadditional geometric objects is not included in the at least onegeometric surface.
 31. The method of claim 1, wherein each of the firstderivatives of the parameterization P_(S) at the corresponding portionagrees with a corresponding each of the first derivatives of theparameterization P_(O) at the corresponding portion.
 32. The method ofclaim 24, wherein each of the one or more additional geometric objectsis continuous.
 33. The method of claim 24, wherein each of the one ormore additional geometric objects is not included in the at least onegeometric surface.
 34. The method of claim 26, wherein each of the oneor more additional geometric objects is continuous.
 35. The method ofclaim 26, wherein each of the one or more additional geometric objectsis not included in the at least one geometric surface.